Can I present myself as a postdoc at a given institution if I have a supervisor, but no official appointment there? I have already finished my PhD at X university. I am now a postdoc at Y university and in collaboration with X university (by "in collaboration with X" I mean that I actually have two supervisors: one from Y and one from X). 

Only Y is paying for me, and, I am currently working in the lab at Y university. But my affiliation is X and Y in the scientific papers. My supervisor at Y accepted that I also use the affiliation X.

As I am using two affiliations X and Y in my scientific papers, can I consider that I am a postdoc at two universities at the same time? Is it possible that I write in my CV that I am a postdoc at two universities even that X is not paying for me?

Any help will be very appreciated.
You may present yourself as a postdoc at X if, and only if, you have an official appointment at X, the kind that comes with an official title, effective start date (and usually an end date), and is documented in some kind of official document of the institution X - a contract, appointment letter, etc, signed by some official person. Otherwise it is simply not true that you “are a postdoc at X”, so obviously you shouldn’t say that you are.

If you have such an appointment (it sounds like you don’t), it would then be appropriate to list it in your CV. The best practice in that case would be to write your official title rather than “postdoc”, although if you want to clarify that the position is a kind of postdoctoral position, it would be reasonable to add this in square brackets or as a footnote.

Note that some official appointments are unpaid, so whether you are paid or not is not the main point.

Finally, everything I said above also addresses the question of whether it’s appropriate to represent X as an institutional affiliation in your papers (if you don’t have an official appointment, it isn’t; the institution hasn’t given you an appointment, which means they have not endorsed you as someone they are willing to have their name associated with on scientific publications).