Answer

The Seventeenth of Tammuz has gone through various stages throughout history. It was instituted, canceled, and reinstated. In the Shulchan Arukh, Orach Chayim 550:1, it is written that everyone is obligated to fast on the four fasts. On this the Magen Avraham comments (and the Mishnah Berurah also cites him) that according to the strict law of the Talmud there is no obligation except on the Ninth of Av, but the Jewish people have already accepted these fasts upon themselves, and one may not breach the fence.
From that wording it seems that this is a custom (about which it is usually said: “one who breaches a fence will be bitten by a snake”). There is a halakhic obligation to observe customs, derived from “Do not forsake your mother’s teaching,” and some say also on the basis of a vow.
The question is not how people feel, but what they actually are. Someone who breaches this or that custom—and even someone who violates an actual law—can still be called “religious” (although that is a sociological term, not a halakhic one. It has no sharp meaning). Perhaps it depends on why he breaches it: if it is because of his impulse, but he understands that he is not acting properly, or because in his view it is permitted—he can still be called religious. But if it is simply because he is not committed, then not. Even if he thinks (mistakenly) that a custom is not binding, that is an error that still places him within the above-mentioned category of religious.
Releasing a custom from obligation applies only where the circumstances have changed or there is some special reason to permit it. At present I do not see such a reason. True, customs contain an inherent paradox (I heard this from Rabbi Eliyahu Blumentzweig of Yerucham): it is a command that instructs us to preserve and not deviate from the innovation. After all, every custom begins as a deviation from what existed before, and preserving the custom means fixing that deviation in place. Therefore, if a different custom begins, even if the first people to practice it are sinners, it is possible that this could create a new binding custom.
Just a technical note. In the present case, though, that is actually not the situation, since the Jewish people fasted all along because it was an obligation. And from the moment the obligation ceased, they continued fasting by force of custom. If so, in this case the custom did not actually change the existing situation.