According to this Dragon+ article, the term tabaxi originates with the race of cat people (at least in the Forgotten Realms setting as of D&D 5th edition):
Tabaxi with a capital T is the ancestral name of one of a number of human tribes that traveled east to Faerûn from across the ocean...
...although having both tabaxi and Tabaxi in the canon of the Forgotten Realms is confusing, it makes sense for tabaxi to originate from Maztica. This provides the in-world explanation for the similarity of their names: the human Tabaxi named their tribe after the mysterious cat people of Maztica and Katashaka. Also, establishing them as native to that distant land explains why tabaxi are rare across Faerûn in both the past and present.
But nothing further is explained about the name beyond that. I haven't found any other information about the name of the race, but I am curious what other information there is, e.g. whether the word "tabaxi" ultimately originates from the cat people, and whether it has any meaning other than "this specific race of cat people".
Likewise, it seems as if the tabaxi should have their own name for their figure of worship, aside from the plain phrase "Cat Lord" in Common. This answer explains that the first Cat Lord in D&D was named Meerclar, but that pre-dates the creation of the tabaxi and the Fiend Folio doesn't mention any figure of worship. (and the 2nd edition books seem to indicate that tabaxi religion is largely animistic.) Likewise, it's mentioned that the corresponding figure in Greyhawk is named Rexfelis, but there don't seem to be cat people in Greyhawk and the Latinate name seems like a poor fit for the tabaxi.
One might presume that "tabaxi" means something like "cat people", and that the Cat Lord could have some related name in the tabaxi's language, but is there any established information about these subjects in any official D&D publications?