Why don't journal publishers ask for proof of affiliation?
The crucial point is: Why would the author lie? Let's try some hypothetical answers:


To bluff the editors and reviewers so that they think you're at a top place. But reviewers will likely be from your field of study and realize that you lie. They would probably know it if you moved to a high-ranking institution.
To make the paper look good in your CV. This is a non-sense, you sell your affiliation in different ways than by listing them in your papers.


I can't think of any other reason. Given that the authors have no incentive to list a false affiliation, there is no reason to verify it.

Also, remember that some affiliations are very hard to verify, for instance if you stay somewhere for 6 months and want to list it as an affiliation, you possibly do not appear in any official lists.