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: 
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.
For the most part, your name and affiliation are not relevant to the content of a paper, which is what a journal is interested in. In the vast majority of cases an author would not have any incentive to lie about such things, so a journal would probably be willing to either take you at your word or only perform some basic checks, unless there were circumstances which aroused suspicion. The only incentive I can think of for an author to disguise their name or affiliation is if they wished to hide a conflict of interest or bad reputation, which I have seen happen. I expect it's pretty rare though.