As I understand it SE did consider such an idea back during ARR's early development but only got around to implementing the Companion system with the default Company Chocobo, as they probaly considered expanding it to all mounts as utterly unworkable (especially as some mounts are literally nothing more than non-autonomous or sentinent vehicles, or mindless animals).

Sadly I don't think SE are going to do anyhting more with the Companion System, as the GC Squadron and Trusts have all but replaced it.