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.
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.