I have heard Rabbi Asher Meza say that if one "adheres to Torah," this makes one a Jew; is this true at all?
Nonsense if taken literally. That is, if one takes it as meaning that a non-Jew becomes a Jew by "adhering" to the Torah without undergoing a full conversion before a bais din in the proper manner. On the other hand if one takes it as a philosophical non-literal meaning, then it can be understood. In ancient times, before Matan Torah, the situation may have been different. That is, one could consider that the wives of the shvatim "converted" as part of the marriage.
Actually, one could conceivably say that "adhering" to the Torah is what a non-Jew does to convince the bais din that he (or she) is serious about conversion and that they should be accepted. Additionally, one could say that this was what was done in the time of King Shlomo when converts were not accepted in order to convince the bais din to accept one as a convert in any event.
One must be careful to understand the meaning of a statement made for hashkafah because misundertanding can lead to bad results. The phrase "grafting onto the tree" is often used by xian missionaries to fool the ignorant.
R' Asher Meza is working off of midrashim and a bit of logic for this argument. He and everyone else agrees that a valid conversion is needed before a gentile can become a Jew; however, he is making a more, over-arching point about the purpose of being a Jew. It's not about who your mother was but rather it's about Torah u'mitzvot hukim u'mishpotim. Now, part of Torah is keeping halacha, and part of halacha is rabbinic in origin and even those halacha that we understand to be d'oreita (from the Torah), we follow only about how Chazal interprets them. So following Torah means following what Chazal said and Chazal said that you can only be a Jew if you convert or if your mother was Jewish. So, if following Torah makes someone a Jew, and Torah means that you have to convert; then, by deduction, to be a Jew, you must convert.
sabbahillel's answer is also pretty good to (especially about the Matan Torah part).