My version of the story of universities and Unix source code
In the beginning there was Research Unix Version 7 from Bell Labs (a part of AT&T). AT&T gave V7 away basically for free to universities and V7 didn't so much come with source code as intrinsically require that source code in order for you to install and operate it sensibly. On top of that V7 needed plenty of hacking to add features to it, so people did. As a result of this, plenty of universities acquired AT&T licenses for V7 (source code included).
When the UCB CSRG created BSD Unix from V7 they gave it away for free because they were, after all, an academic research organization. You had to have a V7 source license from AT&T because the BSDs contained AT&T code (which would, a decade later, contribute to a nasty set of lawsuits). 4.2 BSD was better than V7 but that didn't make fully complete, so lots of people at universities hacked on it, fixed things, and sometimes did really odd things to it in order to support lots of nosy students on inadequate hardware. Like V7 before it, BSD was not infrequently just the raw material for a local computing environment rather than a 'ready for production' boxed pack of software.
(Early Unix was an environment of system programmers, especially once you started running into bugs and swapping bugfixes with other people.)
Soon various Unix vendors got into the action, licensing V7 from AT&T and adding modifications to the BSD base (including porting it to their own hardware and so on). When they sold the result to universities, the university system programmers asked the vendors for the source code because of course the system programmers were going to have to customize the vendor system. For various reasons early Unix vendors said 'sure, if you have an AT&T V7 license because this still descends from V7'. In practice source code was less and less necessary over time and less and less used as vendors successfully turned Unix into something you didn't need system programmers to run. Still, universities kept asking for source out of habit and just because and vendors generally kept giving it to them.
(Or at least old departments and groups with a habit of this kept asking for source code. As Unix expanded into new groups and departments, the new people often didn't ask because they had no use for it.)
There were two complications with this picture. The first one is that at some point AT&T stopped giving out V7 licenses, even to universities (and I think this was fairly early); if your university hadn't gotten into the Unix game early enough and thus was unlucky enough to lack a V7 license, no one would or could give you any further source code. The second is that many vendors themselves started to become increasingly reluctant to give out their source code, making it harder to get or simply no longer making it available at all. My vague impression is that by the late 1990s, even universities were no longer getting source access (right when the free Unixes were coming along to make the whole issue moot, although I think that's just coincidence).
(These days the V7 source code is publicly available at the Unix Heritage Society. I have no idea what this does to assorted licensing issues.)