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 several answers here.

1. 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.
2. 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)
3. 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?)
4. 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?
A journal is interested in whether the submitted article is,

* Appropriate in subject matter for that journal
* Of a suitable standard for that journal (as advised by reviewers)

In an ideal world, neither of these things is indicated by affiliation - so why should they care?