Unix is an aesthetic. It's squishy. Emacs is a text editor. Editing text is very much a Unix Thing. Emacs does a lot of other crap too, most of it (email, terminal emulation, IRC...) intended to replace or augment some other core Unix Thing. Is putting all those Unix Things into a single box (that isn't unix) a Unix Thing? Probably not, but most of us forgive emacs that transgression. Don't use the crap that doesn't fit your world view, and it remains a unix tool.
Emacs predates the Lisp machines, btw. The original was written (as TECO macros) on ITS. The Lisp machines came out of the same culture and thus had their own emacsen. But GNU Emacs that we are all using today was written on Unix from day one.