Unix was initially developed on and for DEC PDP-11 mini-computers, not mainframes, at AT&T.
The problem with Unix, from my experience way back, is that each vendor had their own "mix", or flavor. One company's mix was mostly AT&T System V with a little BSD thrown in, another was mostly BSD with a little System V in the mix, etc. Since the name was trademarked, I think by AT&T, each vendor had their own proprietary name: Silicon Graphics (remember them?) called theirs IRIS/Irix, HP called theirs HP-UX, IBM's was AIX, etc. Very fragmented, and each supported a different set of shells (C shell, Bourne, etc.). Hey, but it was "Unix"!
Oh, and not to forget Solaris, from Sun.