I am genuinely interested, whether Jewish people identify as an ethnicity / race, or as "religious people" (of any race, i.e. as "Christians": not tied to any particular race).
If a non-Jewish man becomes a believer in Judaism, will he then become a "Jew"?
Both and a third option as a nationality.
It is an ethnicity in the sense that being born or raised in a Jewish community comes with a culture attached; a community-specific language-variant, shared historical and cultural symbols, food and norms. What exactly makes 'ethnicity' is difficult to define, but if we define it as a grouping of people who identify with each other based on ingroup-destinguishing attributes such as traditions, ancestry, language, history, culture, religion, or social treatment then Jews as a people check of all these boxes.
Jewish ethnicity is also defined by its religion, in many ways because Jewish religion is community centred. Jewish religion is designed to be practice communally, and is not what many secular people now often think of religion, namely an individual philosophy you live behind your front door. Jewish religion shapes all aspects of an observant Jew's life from the food that is eaten to the way shoelaces are tied.
You can be part of the Jewish ethnicity without yourself being religious, because the religion has shaped the ethnicity. You can speak Jewish lingo, grow up with Jewish food, music and history without being religious. But religion is an important aspect of being a religious Jew and being one means you become part of a sub-ethnicity within the larger Jewish ethnicity.
Jews are a nation, because not only were we a literal nation with a King and whatnot, but entry to the Jewish religion also means entry to the Jewish ethnicity (at least in Jewish eyes). This is more than just 'religion'. Some religions evangelize and deem it important to convert the world. Jewish religion does not view the world this way. In Jewish thought, every non-jewish 'nation' has its own intrinsic value and mission in the world. It is not an inherently good thing for non-jews to become jewish. To become Jewish you have to demonstrate a willingness to be part of the Jewish people.
To become Jewish is similar in many ways to how one becomes a Dutch or US citizen: you apply for citizenship, you demonstrate that you have knowledge of important things that would make you an effective citizens, such as important laws or cultural norms, and as soon as you are okayed, you get your passport and from that point onward you are a citizen of that nation. This means that all laws applied to its citizens now also apply to you. So transgressing laws of kashrut, such as eating cheeseburgers, will now be put on your record. The main difference with Jews is that your Jewish citenship is applied retroactively.
Note (to answer one of your specific questions): merely adopting Jewish religion is not sufficient to become a Jew. You still have to apply for citizenship so to say. How the conversion process works is very time and community specific, but generally you approach a Rabbi, actively live in a Jewish community, learn Judaism and eventually, if it is decided (by the relevant religious authorities) that you are ready to become a sincere observant Jew, than your citizenship will be made official. Browse this site for excellent questions and answers going into details about this process.
So, to summarize. Yes.