Do postdoctoral studies have official certificate?

Certainly not in the US. I never got any kind of certificate for the postdoc I did, and in fact I'd never heard of such a thing until your question.

My current employer never asked for any formal verification of my postdoc. They received letters of recommendation from my postdoc supervisor and other colleagues at that institution, and I think most US academic employers would consider that sufficient.

Then again, they never asked for my PhD diploma either. In the US, we don't use the diploma for official purposes; it's just a decorative piece of paper. The official document of choice is a transcript sent directly from the university (and my employer did ask for one of those).

I'm in the UK. I've never heard of a "certificate of postdoctoral studies", not least because postdocs aren't studying towards a qualification: they're working. I've also never heard of a postdoc not having a formal contract of employment. In fact, I suspect it would be illegal to do so in the UK (and probably most western countries).

You are expected to have publications while working as a post doc. These publications document your institution, topic and success while working as a post doc.

You can also ask for recommendation. I do not think it is a separate "diploma" because unlike PhD, post doc is already more work than studies.