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.
There are some examples of papers published under false names or pseudonyms. For example, [Student's t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution#History_and_etymology). A possible scenario is a scientist working in a private institution which doesn't allow him to legally disclose his research. Just like books can be published under pen names, scientific articles can be written using pseudonyms (see [If I publish under a pseudonym, can I still take credit for my work?](https://academia.stackexchange.com/q/8603/6315)). If that's allowed, it makes no sense to check credentials, including affiliation.