What to do when PhD approval takes so long? I am a PhD student in mathematics in Israel. I still have a couple of years until I graduate, so I am asking this question mostly out of curiosity.

I've recently learned that at least in my university, and at least in mathematics, there is no thesis defense for PhD students. The approval process for a PhD dissertation is very different than the process in the US: Once your advisor decides the thesis is complete, you submit it to the university. A committee (which does NOT include your advisor) then chooses two non-Israeli reviewers, send them the written manuscript and ask them to review it (anonymously). The reviewers send their reports, and then the committee decides whether to approve the thesis. Finally, the university senate should vote to approve the committee's decision, but this is basically a formality.

The main point is that this process is a long one. Once the advisor approves the thesis and it is submitted, the student is no longer considered a PhD student and stops receiving a fellowship. For this reason, most students submit their work in August, near the end of the academic year. However, it can easily take 6 more months for the PhD to be formally approved - the reviewers can pretty much take as long as they like to review the work, and it also takes time for the two committees to convene.

I was wondering if such a process exists in other countries as well. In particular, are American universities aware of the fact that sometimes a PhD graduate will not actually hold a PhD for many months after beginning his postdoc? Does this cause any problems? 

Since many Israeli students graduate every year, and many of them find a postdoc in elite American universities, I assume there is no real problem. But I am curious as to how this process works in practice. Will universities in the US let you teach classes and hold the title "Professor" without a formal approval of your PhD?
While the details of the assessment/examination process vary around the world, everywhere (that I'm aware of) has the same issue that there is a time lag between handing in your thesis and receiving a decision. That time lag is a few weeks as a minimum, often several months.

This does not generally cause major issues with job applications. Employers who are accustomed to hiring postdoctoral researchers will be very familiar with the delay, and are generally happy to offer a job and have you start working (and being paid) pending the award of the PhD. It will be clear to them, from the reference/recommendation letter from your advisor, whether or not you are likely to pass the PhD. If there is a chance that you might fail completely, it's unlikely that you would even be applying for postdoc jobs, let alone being offered them. I think that technically the employer would have the option of reconsidering your contract were you to unexpectedly fail the PhD, but I have never heard of that actually occurring in practice.