“Rabbi” is poorly defined. Arguably is acts more as a courtesy title such as “Mister” in English, or “Monsieur” in French (both of which once indicated the you were addressing person higher up on the feudal ladder than a mere peasant). Academically, it seems to indicate you can teach older children (meaning it is similar to a Bachelor’s degree, or in modern times with watering down of credential, a Master’s).
“Admor” (literally, My Lord and Teacher and Rabbi”, and “Lord” is in the fedual English meaning showing respect to someone whom you acknowledge outranks you substantially) implies you are accepting the person as a leader.
As to whether it is inherited, one should note that in most traditional cultures, our’s included, positions of leadership will tend to stay in a family, with flexibility to get rid of the unworthy and raise up the worthy. An OTD from an hashuvah (“important”, though “noble” is a better translation) family is seen as having died a horrible death and ignored, and a BT who distinguishes himself gets to be the distinguished ancestor from whom people are proud to trace descent (so in many ways we are more like the Americans, than the English in such matters, even though we are still a very non-republican (small “r”) culture).
Note that among Americans, names such as Brown of California, Cuomo of New York, Kennedy, Bush, etc., appear to end up in positions important based on inheriting from their parents (so while the son of a leader isn’t necessarily a leader, it sure helps).