Lisp Machines were never that much in direct competition with UNIX machines as such. They were competing with UNIX + LISP. Research labs who programmed in Lisp bought Unix systems plus a Lisp environment (like Allegro CL) or bought a Lisp system (like Allegro CL) for their existing Unix environment. But Unix systems were used for quite different things than Lisp programming or running Lisp apps.

When Lisp Machines lost in the market, the Lisp users moved to UNIX + Lisp. Companies like Franz and Harlequin/LispWorks that came from that market still exist today.

Lisp was always used on different systems. When Lisp Machines were 'popular' (we are talking about less then ten thousand machines that were ever build and sold), Lisp was used on Windows, Macs, Mainframes, Unix and other operating systems.

I should also mention that almost all Lisp Machines were single user personal workstations - initially specially aimed mostly at AI programmers.