There’s a Wikipedia entry on this, where it says:
An expression of Jewish nationhood
The characteristics of the State of Israel, as well as the Declaration of Independence, identify it as a “Jewish nation-state,” that is, a state influenced by its connection to Jewish tradition and heritage. The Jewish character of the State of Israel is expressed in four components.[1]

The first component: Zionism, in its original and most basic sense, was a national liberation movement that sought to free the Jews from dependence on external factors in determining their fate, and it aspired to attain sovereign means including political, military, and economic power that would allow the Jews to define themselves and their destiny.

The second component that defines Israel as a Jewish state is the Law of Return. This law, closely tied to the first component of national self-determination, establishes that all Jews, wherever they are, have a right to citizenship in the State of Israel, and they can make the State of Israel their home if they so desire. Israel was established, among other reasons, to prevent a situation in which Jews seeking refuge would be knocking on the doors of countries that do not want them present. The justification for the Law of Return in Israel does not depend only on arguments of possible distress. If ethno-cultural groups have a right to self-determination—that is, the right to maintain a sovereign space in which they are the majority of the population and in which their culture develops and flourishes.

The third component defining Israel as a Jewish state concerns the public sphere—such as the state’s symbols, its official language, and its calendar. These features are connected to the content of Jewish cultural and traditional symbols, such as the menorah, the Star of David, and the Hebrew language. The state calendar is shaped according to the Hebrew calendar, and the Sabbath and Jewish holidays are the official days of rest.

The fourth and most important component is connected to the public education system. In the State of Israel, as a Jewish state, the public education system is supposed to be committed to the continuity and flourishing of Jewish cultures—Jewish cultures in the plural, because within the Jewish nation there are completely different understandings of the character of Jewish life and the meaning of Jewish education.

In the State of Israel, many laws have been enacted with Jewish characteristics. These laws are divided into three main categories:

Laws that stem from its being the state of the Jewish people (for example: Basic Law: Israel – The Nation-State of the Jewish People, the Law of Return, the Citizenship Law, the Law on the Status of the World Zionist Organization – The Jewish Agency for the Land of Israel)
Laws that stem from Jewish law (for example: the Matzah Holiday Law, the Rabbinical Court Law, the Prohibition on Raising Pigs Law)
Laws that stem from Jewish culture (for example: the Broadcasting Authority Law, the Foundations of Law Act, the State Education Law).
The emphasis on the state’s Jewish character, at the expense of its democratic character, is reflected in public debate on a variety of issues such as opening businesses on the Sabbath, the authority of the rabbinical courts, and evacuation of territories in peace agreements.
…
Justice Barak ruled in the judgment that the most minimal interpretation of the term “Jewish state” is as follows:

“At the center stands the right of every Jew to immigrate to the State of Israel, in which the Jews will constitute a majority; Hebrew is the state’s principal official language, and most of its holidays and symbols reflect the national revival of the Jewish people; the heritage of Israel is a central component of its religious and cultural heritage.”

Exactly what I said: ethnicity without value-components.

I’ll take the liberty of answering for myself that first of all they were presumably talking about (as the Rabbi said) “the state of the Jews” as opposed to a state of all its citizens. And then there is practical significance to several things. For example: 1. Why would anyone at all (a Jew and a sane person) risk his life for a state that is not his, but merely a contract among people who have no connection to him? 2. A concept of ownership of the geographic territory of the state by the people (the nation). That is, the relation between each individual’s ownership of land in such a state and the ownership of the people over all the lands of the state (the people’s ownership of its land) is roughly like the relation between the ownership of someone renting a room in another person’s apartment, over that room, and the ownership of the landlord over that room. And I’m speaking especially about the Arab citizens, who do not belong to the Jewish nation to which the state belongs (and which, according to the idea of a state of all its citizens, does not exist at all).

Even if all that is true, there are no values here, only ethnicity. But on the substance I really don’t agree. Regardless of nationhood, if I am a citizen of a state and it is threatened, I defend it. That is how it is in every state, regardless of national cohesion.
As for ownership too, all the citizens together are a group that can also have sovereignty.

How the mighty have fallen

Of course there are no values here. It’s just the simple (primitive) life reality of every people. But there’s no need for values at all. It’s like if there’s a family, they buy an apartment. The state is a necessary instrument (the regulation of the formal aspects of the relationships among the people in society). But just as a legal system doesn’t replace society’s natural sense of justice, so too people did not in practice simply gather in order to establish a state (somewhere in the world in history). There always was (and needs to be) some bond among the people that is more than transactional, and for that reason there really is no state in the world that is a state of all its citizens, not even the USA (they constantly talk about the American people or nation). If there were no American people, but only a business association, I doubt anyone would go out to risk his life in the mountains of Afghanistan. I refer the Rabbi to what he said on this in “Two Carts.” Citizens and a state are tools that serve an essence—a people, or at least some bond that is more than a business contract. They cannot become the essence itself. In any case, I wouldn’t rely in battle on someone who sees me as nothing more than a fellow citizen. Form does not replace content and essence. Law does not replace justice, and for a state to be stable it cannot replace a people. That is also the reason for America’s internal disintegration today. Half of what used to be the people there doesn’t understand this point. And so too with a third of the people in Israel.

Indeed, how? All I claimed is that there are no values here, only national-ethnic identity and nothing more. I’m not criticizing that, and I completely agree that there doesn’t need to be anything beyond it. The implications regarding America and its disintegration are far from the truth—as far as the fallen are from the mighty.
And by the way, Americans are definitely willing to die for America even today. I wouldn’t recommend that anyone declare war on them. What they are tired of is dying for others and being the world’s policeman, and rightly so.

Our mourning has been turned into a violin, our lament into a dance, and our sackcloth into joy. And by the way to your by the way, it seems to me that it’s only the Republican half that is willing to die for America. But the Democrats, who in practice are the majority (though they achieved that majority by taking in immigrants who, as long as they haven’t assimilated into American culture, are still not part of the American people), are tirelessly undermining the very concept of a people through the steamrollers of political correctness, and the reaction to that in Trump’s election is temporary. Therefore my prophecy is that America will disintegrate in the near future, and I hope I’m proven wrong. And I too justify and support their not dying for others. (And it seems to me that in the past I also justified their non-intervention to the Rabbi in a case where he himself did not agree with that non-intervention.) As long as no one cheats and lies, every nation should look after its own interests.

Following what you wrote above, that “there is no such thing as a Jewish state except perhaps in a national sense (which itself is empty of any value-content, aside from the ethnicity of the residents).” Elsewhere you wrote that the concept “Jew” is a halakhic concept. And Justice Barak wrote regarding the interpretation of the term “Jewish state”: “At the center stands the right of every Jew to immigrate to the State of Israel.” According to this, there is value-halakhic content to the Jewishness of the state in the sense that anyone considered Jewish according to Jewish law is entitled to immigrate to Israel. Beyond that, do you think it is appropriate that the state be a Jewish state? And if so, in what sense?

That is not value-content but ethnic content. It is a state that exists for Jews, and therefore only they are entitled to immigrate to it. It’s like talking about the value dimension in a club for men who are 1.78 meters tall, where only such men are allowed to enter. Beyond that, I ask: who are those “Jews” who are allowed to immigrate? What is their definition? This is an ethnic definition alone, without any value dimension and not even a cultural one.
It reminds me of something I once wondered about regarding Rabbi Yehuda HaNasi’s view that lesser holy offerings are the property of their owners. I wondered in what sense they are the owner’s property, since they have to be offered as sacrifices. To that they answered me that it’s in the sense that one can betroth a woman with them. But that is of course not an answer. The possibility of betrothal is a result of the fact that they are the owner’s property, but I am asking in what sense they are his property, not what the implications of that determination are. This is just an example, and obviously much more could be said about it.

But if the above club were intended for men who are 1.78 meters tall or for anyone who donated a kidney in his lifetime, wouldn’t that count as some kind of value dimension of the club? (Kidney donation is analogous to someone who converted according to Jewish law, which is an act with halakhic value significance.) One could also say that the men who are 1.78 meters tall must be descendants of someone who donated a kidney in his lifetime (another value aspect of the club). It would then come out that this club is a club for kidney donors and their descendants who are 1.78 meters tall.

Indeed. And that is exactly what I wrote: in the essential sense there is nothing here beyond ethnicity (because Jewishness itself, according to their view, does not include values, only the implication of permitting their arrival in the country—their entry into the club), and therefore it is exactly like the example of men who are 1.78 meters tall. If we were speaking about a Jew as someone who donated a kidney, or someone committed to value A or B, that would introduce substantive content. Without that, there is only ethnicity.

But even according to their view, a Jew is someone who converted halakhically (accepted upon himself the yoke of Torah and commandments = performed a value-laden act, like someone who donated a kidney) or a descendant of someone who converted halakhically (whether he literally converted or accepted the Torah at Mount Sinai, which is a kind of conversion). If this were only about ethnicity, then someone who is not descended from a Jew should not be able to enter the club (even if he converted and accepted upon himself the yoke of Torah and commandments). Besides that, from your words I didn’t understand whether you want the state to remain a Jewish state, or whether it makes no difference because in any case it lacks value significance.

According to their view, acceptance of the commandments is incidental and arbitrary. There is nothing essential in it, and certainly nothing of value. It’s just that in the past Judaism was defined in value terms, but in their view that is a malfunction. So today they will tell you that a Jew is whoever defines himself as a Jew, and no conversion according to Jewish law is needed. It’s like a club for the descendants of kidney donors who don’t think that donating a kidney is a good thing.
I’d be glad for it to remain Jewish, because I too feel comfortable living among my own people (even if not among my own religion). But I don’t see that as a value, only as a need. I also want the state to prosper economically, because that too is our need.