Why was Chidi writing a thesis?

This is more of a treatment of whether Chidi has a PhD or not, so it is indeed a cross-over with the academia SE.

He is probably revising his PhD manuscript for a book publication.

In social sciences, PhD theses are usually published as books and the more respectable the publisher the harder it is to have your theses published. Similar practices might be occurring in natural sciences as well, but I haven't heard anyone publishing their entire theses as books. (They are usually divided and pieces are published as book chapters or journal articles.)

Most researchers take some time after being awarded their degree to revise their thesis as a book. I guess Chidi just take forever in the revision process and his thesis became his torture, which Michael uses against him.

Most PhD courses have page or word limit for their dissertation.

Chidi is a caricature character with many exaggerated traits. The length of his thesis is no different. In order to save the thesis adviser and the degree committee from the pain Chidi inflicts on his adviser, a page limit is designed. As a side information, some institutions have lower page limit as well but there are a few great theses with a few pages, most notably John Nash's 26 page and 2 reference thesis.

Going back to Chidi, there is a good chance he submitted a shorter version to get his degree and kept perfecting it until he died for a book proposal/submission.

Chidi acts as PI in their project

PI, principal investigator, is almost always a PhD. Some institutions don't even allow PhDs, they have another position called senior post-doctorate researcher who can be PIs, as opposed to junior PDRs who cannot. People can apply for grants before they are awarded their title, but it is usually granted on the condition that they receive their PhD.

Some university lecturers like to be called professors and some might have large rooms

Chidi is called professor of philosophy which also shows that he has a PhD. Even though all evidence points to Chidi having a PhD, being called professor is not a conclusive evidence. K Mo's answer gives details about the title of professor in Australia.

Room allocation is usually a dibs process only depending on seniority. After all senior faculty numbers are allocated, if there are rooms available, PhD students might get a spacious room as well, but they usually don't.

Out of Universe details

I believe, the words thesis or dissertation are stronger than paper or book for a regular viewer. If the producers wanted to emphasise on him as an academician, writing a thesis helps more than writing a book.