Do postdoctoral studies have official certificate? I have seen that some universities give postdoctoral fellows an official certificate (similar to the PhD diploma) after finishing their postdoctoral job. On the other hand, some of postdoctoral positions are just simple research (or even academic) jobs. I have even seen that some professors pay the postdoctoral fellows without contract (from their grants).

My question is: How common is a certificate of postdoctoral studies? Do future employers expect applicants to give them certificate of postdoctoral studies as well as PhD diploma?
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).