Is it common to refer to someone as "Prof. Dr. [LastName]"?

German academia traditionally expects that one will use all relevant titles, so Prof. Dr. is pretty common there. Likewise other places with an academic system related to Germany in some way. As you note, in the US this would be very uncommon and the two titles you mention often used interchangeably there, even when it isn't clear that both apply.

And, if I remember correctly, it is always Prof. Dr. and never Dr. Prof. since the professorship was earned after the doctorate and is a "higher level" honorific. But "Herr Prof. Dr." doesn't fit that rule, I guess.

Background. In Italy, Dottore (short version dr. or dott.) refers to those who hold either a bachelor's degree, or a graduate degree, or a Ph.D., or those who are physicians.

Reply. Those who both serve as physicians and teach at medical school are commonly regarded as Prof. Dott. or Prof. Dr.

Yes, it is common to use both, and the reason is that these are two completely different categories. Doctorates are academic degrees that come in different flavors including "Dr. h.c." and "Dr. habil.". They indicate the academic level, just like Bachelor or Master degrees. For academic degrees, you typically use all degrees on the highest level, so you don't mention a Master degree when someone also holds a PhD, but you mention all doctorates. "Professor" on the other hand is (in almost all cases) an official title which you mention just like you would always refer to a judge as "Judge XXX", whether or not he/she's holds a PhD. That also explains the order: Titles come first, and you lose them when you lose the job. Degrees become part of the name (so to say, until recently, it was possible to have them on your ID card.)