Okay To Shop Around?

[Really?]

If we're talking about a book (self-published, no less) whose contents are not a result of a research process judged by peer review (see above), perhaps your use of the term "degrees by published work" deserves a bit more scrutiny. Because that isn't what it means. Not by a long shot.

Peer review? You mean the kind of stuff mentioned here, with the two identical abstracts? PFFT!
I suspect the degree by published work in the uk is pretty well established beyond whatever anyone may insinuate.

Again, a non sequitur. I described a very "established" process and, yes, it is used in the UK.

Peer review does not refer to doctoral dissertations. Those are not reviewed by peers. They are reviewed by faculty. Peer review occurs in academic journals, where one's work is reviewed by other scholars, and authors are considered peers, not students.

Handing someone a Ph.D. for a non-supervised, self-published book however, is not. Neither is doing it for said work after it had been submitted for a previous Ph.D. Not at a university, anyway.