Even before they were routinely interconnected, Unix systems could be considered as effectively their own BBSes. They were usually multi-user systems, and they allowed their users to swap files, and exchange messages (initially as a specific use of shared files). Users didn’t have to connect to a separate BBS.
Interconnecting Unix systems, whether permanently or intermittently (using UUCP), provided an extension to this, and protocols were created to allow users to share files with users on other systems, and to send and receive messages to and from users on other systems. Successive changes to protocols and new protocols provided more and more indirection, for example going from bang paths for email to the present-day DNS-based system.
There was one significant difference between BBSes as seen from micros, and Unix systems: the former tended to favour asynchronous use, since only a small number of users could be connected simultaneously (compared to the overall number of users), whereas the latter also allowed synchronous messaging systems (starting with
write which was already present in V1 Unix) and related programs. This led in particular to the emergence of MUDs (multi-user dungeons) and the communities surrounding them; these were nominally multi-user text-based games, but often users were interested more in chatting with other users than actually playing the game. (MUDs weren’t only available on Unix systems; Micronet in particular in the UK offered one which was accessible over Prestel.)
Many Unix systems also provided dial-up access; this was fairly straightforward since modems could be considered as extensions of the serial lines used to connect terminals to Unix systems. I don’t know of any which had the same access patterns as BBSes; most would have provided dial-up access for regular users working from home (although logins could of course be shared, or given out more widely than the bean-counters intended), and some provided commercial access to specific services.
Many early ISPs effectively provided dial-up connectivity to Unix systems; but the latter weren’t the end-users’ target, they were simply a hop on the way to the rest of the Internet, either directly (for interactive protocols) or indirectly (for email, Usenet etc.).