See Berakhot 34 regarding mentioning the name.
As for familiarity with the object, that's just adding more support—simply as the Rabbi already answered you.

Don't you have his ID number?

I liked the answer. Of course I agree that God can send His blessing even without the full name.
It's just that I feel there are people who are kind of particular about this, and if they don't know the full name then they'll pray significantly less. I think there are people who understand that there is some point to the prayer being specific.

Thank you very much, Moshe, but I seem to remember explicitly that some say all this applies when the person is in front of you; I just don't remember the sources right now. But I did see in Magen Avraham, quoting the Maharil, this distinction regarding Berakhot 34.
P.S. Nice joke.

Indeed, maybe one could also raise a difficulty from two people with the same names, as we see in Gittin: "his name is the same as his name and her name is the same as her name"—and if so, perhaps according to that view the prayer could get mixed up on the way…

Rabbi, you presented two rejected possibilities: either the Holy One, blessed be He, doesn't know without the explicit name, or prayer is a machine. That leaves a third possibility: that God indeed knows even without the explicit name, and prayer is also not a machine (and I understand, superficially, that it is rather like any request to any other person—sometimes God is appeased and sometimes not).
But there is an assumption here that God's knowledge alone is enough for the prayer to "take effect," and that prayer has no validity conditions such that without them it is rejected outright. What is the basis for that assumption? Is it a general intuition that there shouldn't be any threshold conditions or formal parameters, unlike fulfilling obligations in commandments, or is it that in principle such threshold conditions could exist, except that we haven't heard of them?
Does prayer in thought "help"? As a matter of Jewish law one can say that this is the law for various reasons, but for example, why did the prophets in the Hebrew Bible (Tanakh) pray verbally and not silently in thought—like Jonah in the belly of the fish, who made the effort to "say" his prayer: "I called out of my distress to the Lord, and He answered me," etc.?

There's also the possibility that the Holy One, blessed be He, dances the hora, and if He lands on His left foot He performs a miracle. There are lots of possibilities. Think about what sounds reasonable to you.

I thought about it, and the possibility you just added now sounds entirely possible to me too, but it has no basis. The one I presented (threshold conditions, and then acceptance of the prayer according to God's decision) is, with all due distinction and to my understanding, the accepted view (especially if one takes the kabbalists into account), and it also fits with the fact that there is apparently at least one known threshold condition, namely verbal expression, which the prophets used and who presumably knew a thing or two. So in order to reject it, I'd want some counter-argument.

It has a very logical basis. If you can pray that you meet your friend tomorrow morning and you don't need to mention his name, or that your wheat should grow in the field, I don't see any reason in the world why you shouldn't be able to pray for some person to recover without giving an ID number and filling out a form. If you're looking for stronger arguments than that—I don't have any. I would be looking for arguments that you do need to fill out the form. But everyone to his own taste.

So the reasoning is not that the claim that you don't need a name is independently plausible in itself, but that it rests on an analogy from other prayers?

Our master the Rabbi, Full Width, may he live a good long life, are you sure you're serious with these pilpulim?
There's a plain-as-day argument here, and it also has support from other prayers. It's both together.

I freely admit that I am serious about these pilpulim. As far as I'm concerned, God is a rather mysterious sort, and even if I manage to get an intuitive feel for arguments in analytical learning, probably as a result of habit in the discipline, I still find it surprising to see arguments that touch on how, what, how much, and why God conducts Himself. Therefore analogical arguments, where one understands something from the given data (and "recognizes" the generalization directly, etc.), are in a completely different category from independent arguments. But as they say that Yitzhak Shamir used to say: Tov.

And where did the idea come from that names are needed in prayer? Isn't that also based on assumptions about how the Holy One, blessed be He, operates? There you don't need justifications?

It's not something I understand, but people customarily pray using the person's name and mother's name, so I assume you need a positive argument in order to decisively say that you don't need it, unlike the hora-dancing example. A quick Google search also gives Babylonian Talmud Shabbat 66: "all incantations are by the mother's name" (the targets of whispered formulas are not machines), and I remember seeing some kabbalistic fussing about the matter.
And even when there is a positive argument ("plain as day"), I thought there was a big synthetic difference between whether it deals directly with God (God understands the matter and therefore will accept it even without an ID number) or whether it is inferred from the prayers (by analogy on the basis of known prayers and those who pray them—even though assumptions about the Holy One, blessed be He, enter there too). In any case, it's clear that my interest was not in God's conduct regarding names in prayer, and I tried to stay within the local discussion in order to sample your synthetic style and then understand it afterward with my own tools. If not, then not.

See Masoret Moshe, vol. 2, ch. 89 on this issue,
but that's only regarding prayer without the mother's name; I don't know what about prayer without the full name… it may be that there is an additional deficiency there.
https://tablet.otzar.org/pages/?&pagenum=445&book=198793

P.S. I think Otzar is currently open free for books, and Machon Jerusalem too until Rosh Chodesh.