It is possible to obtain two PhD degrees (various fields) on the basis of one interdisciplinary PhD dissertation? Is this something feasible (in Europe)?
Or maybe there must be two different 'physical' doctoral dissertations?
No, at least the regulations that I am aware of require that the dissertation be a novel contribution to the respective field and that it has not been submitted before.
If you write a dissertation at the crossroads of different fields, you have to decide for a field in which you want to formally graduate and obtain a degree. It may be advisable to look for supervisors from different fields or with an interdisciplinary background.
It's hard to imagine any legitimate university allowing a thesis that has already been submitted for some other degree, whether at that university or somewhere else.* If you lied and claimed it was a new thesis, your degree would be revoked if they ever found out. Furthermore, there are more requirements to earning a PhD than just presenting a dissertation. You're not going to be able to register as a student and submit a pre-prepared dissertation the next day.
And, anyway, what's the point? A PhD is generally treated as a level of achievement, rather than a quantity of achievement. Getting two PhDs is like getting two driving licenses. That doesn't show you're twice as good a driver – it's just saying "I can drive" twice. Why would you want to prove twice that you're at the same level, instead of using all that time to advance beyond the level of being a grad student?
Having two PhDs is so unusual that everybody is going to ask you about both of them. How impressed do you think they'll be when you admit that this second PhD you were bragging about was just the resubmission of the same dissertation to another university?
* And note that la femme cosmique's answer is a case of the same thesis being simultaneously submitted to two universities as part of a joint programme that both universities agree to when the student enrolled.
It is possible to do this, but usually the arrangements are pre-designated before you do the PhD. My answer is going to be very specific here; I have a friend who did this, and I know it's "a thing" at least in France. But as I said, the scope of the answer is therefore limited to a PhD done jointly between a French lab and one somewhere else.
A friend of mine has done a "co-tutelle internationale de these" which means that he spent 1 year of his PhD in his home country (China) and 2 years in a lab in France (paid by the Chinese government). At the end he wrote his thesis in Chinese and again in English (with a French abstract), defended in China, and he has two PhDs; one from the home institution and one from the French one. These agreements are somewhat common, at least in France, although I cannot really speak for other countries. However, that does mean that it is theoretically possible. As far as I'm aware, besides the language, the theses he submitted were the same.
It is not difficult to imagine a situation in which one of the labs is a different discipline to another (e.g. something like engineering vs. astrophysics) and so you could end up with 2 PhDs in such an arrangement. However these things are pre-arranged before the start of the PhD, and to form a new one would probably be a lot of administrative work.
However, it's possible, at least here.
Some stuff about it is written here:
Especially the paragraph that I'll poorly translate:
Which degree do we obtain?
Each cotutelle thesis is held within the framework of a convention linking two institutions of which one is necessarily French. The procedures and rules are those of the French doctorate and those of the doctorate in the partner country. The two universities recognize the validity of the cotutelle set up and that of the degree supported (grade of Doctor for French university and equivalent diploma for foreign university).
Concerning the issue of the diploma, there are 2 possibilities:
• The student receives a Doctor's degree conferred jointly by both institutions. The diploma is mentioned under his two appellations (for example: PhD in French literature and PhD in French literature).
• The student receives two doctoral degrees from each institution. Each diploma then bears the mention of the diploma specific to each institution, mentions the fact that the thesis was made in co-supervision and specifies the name of the partner institution.
In both cases, the thesis is defended in only one of the two institutions associated with the cotutelle, by decision of the two research directors.
But this is a highly-specific answer which is France-centric. Is that possible elsewhere? Probably. But I don't know enough about it to answer.
It depends on what you mean by two PhD degrees. The University of Michigan allows dual PhD degrees, for example in Physics (or a bunch of other subjects) and Scientific Computing. There is no possibility for a stand-alone degree in the latter but after you fulfill certain requirements, you get a PhD in Your-Subject and Scientific Computing. The requirements comprise a certain units of classes and that the student "make extensive use of large-scale computation, computational methods, or algorithms for advanced computer architectures in their doctoral studies".