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.
Because academia used to be about science, scholarship, and the advancement of learning and predominantly practiced by folk who would not dream of fibbing about such matters.
Because editors have busy lives and they are not the police and journals are not official organs of the state.
Because universities would only care if something controversial or harmful etc was unauthorisedly stated under their aegis - but they do not mind getting the credit for solid work.
Because if some third party does object (hey! someone has read my paper, or at least the author list!), the affiliation is easily checked with the uni (a quick call, a visit to a web site) and the matter resolved.
Because there are various entirely valid reasons for a person to be affiliated with the uni for the purposes of the paper, even when they are not (no longer, not yet) at that uni right now.