Question: It seems like journal publishers do not require credentials for proof of your affiliation and identity. There are many places where such credentials are important, but even the most popular journal publishers do not ask for them. Why is that so?

Answer: 
The crucial point is: Why would the author lie? Let's try some hypothetical answers:

1. **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.
2. **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.
A journal is interested in whether the submitted article is,

* Appropriate in subject matter for that journal
* Of a suitable standard for that journal (as advised by reviewers)

In an ideal world, neither of these things is indicated by affiliation - so why should they care?