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?
There are several answers here.
They quite probably already do so (eg if a claimed affiliation to a prestigious institution looks too good to be true, or unlikely given other information, or a reviewer says "hey, wait..."), but on an informal and ad-hoc basis, rather than doing it for the 99% of unremarkable cases.
They usually don't need to. Most submitters will provide an institutional email address, which is itself a fairly good indication that you are affiliated with that institution. (In the old days, you might have used letterhead - same sort of thing)
Beyond this, defining "credentials" would be complex. Would you need them to point to an institutional webpage with their name on? Submit a payslip? Produce a certificate of employment? (And what would you define as "counting" for affiliation?)
Finally (and most importantly) most of the publishing system is based on trust. The publisher trusts you to have actually carried out the experiments, and to have reported them honestly and comprehensively. They trust you not to have plagarised, or committed ethical breaches, or misrepresented other researchers. They may ask you to sign something to certify you've done all these things correctly, but they won't ask for evidence that someone else has verified you did them. If they're willing to take your word on the actual content of your science, why be particularly distrustful of your affiliation?
The crucial point is: Why would the author lie? Let's try some hypothetical answers:
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.
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. 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?). 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.
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.