So I recall somewhere (probably in the Gemara) that if a person who is called by a certain name for +30 days, that name gets to be his new name, is that true? I know the sages always said stuff that has really important meaning behind it, so is this a deep meaning thing, something that we used to do, or literal thing that we still do?
I think you're thinking of Bava Basra 167b. In context, the Gemara is questioning whether the scribe needs to know the man's and/or woman's names on bills of divorce or receipt ahead of time. The concern is that there might be two men with the same name in the city, or two women with the same name in the city, and the man may give it to a woman who's not actually his wife, she will think she's divorced when she isn't, and she'll remarry while still being married already. In that discussion, the Gemara asks:
וליחוש דלמא אזיל למתא אחריתא ומחזיק ליה לשמיה ביוסף בן שמעון וכתיב גיטא וממטי ליה לאיתתיה דהיאך
But we should be concerned that maybe he'll go to another city, establish a false name for himself as Yosef ben Shimon, write the bill of divorce and give it to the wife of another man [whose name actually is Yosef ben Shimon]!
The Gemara answers:
אמר להו רב הונא בר חיננא הכי אמר רב כל שהוחזק שמו בעיר שלשים יום אין חוששין לו
Rav Huna bar Chinana said to them, thus did Rav say: Anyone whose name has been established in a city for thirty days, there's no concern about him [that the name is false].
If this is indeed what you're thinking of, then it doesn't sound like this is something that was regularly done, so much as a theoretical case that they were addressing. When you ask it there's a "deep meaning" or if it's "something that we used to do, or literal thing we still do," the answer seems to be no to each of those questions.