How did Milton Babitt teach math at Princeton and research math, without an undergraduate degree in math?
The first part of the answer is that in the 1940s, requirements for hiring at American universities were clearly different than they are now. American universities at the time had not developed the world-leading position they have now, and hiring decisions may have had something to do with it -- though it also goes the other way around: If you're not at the level of Harvard today, you need not shoot so high in your hiring; you may also not be able to attract the same level of talent.

Secondly, the early 1940s were particularly difficult, with a large number of talented people needed elsewhere in the nation and not available to hire at universities. In addition, many others needed to get an education and so the demand for teachers was particularly high -- formal qualifications be damned.

Thirdly, in reality, universities have always (and continue) hired people who are qualified, but degrees do not equal qualification. It is generally true that qualified math professor applicants will have a PhD today, but that is not universally true and universities can and do hire people who don't have one if a candidate has demonstrated exceptional qualification in other ways. Babbitt may have been able to show that he knows what he was talking about in ways that convinced those who hired him that insisting on some kind of degree or title was pointless.