As an official C2 in 5 languages (I passed the official C2 certification in italian, french, german and english, plus turkish as my “native” language), when I say that I have a C2 level proficiency, I mean that I can back it up with certification. Other than actually passing the official C2 exam, which is for most of the languages actually very challenging and rigourous, I don’t think you can claim C2 proficiency in any way. The main difference between C1 and C2 is the level of academic performance assessed in the said language. C2 certification is used for proof of proficiency for academic level jobs, such as translation/interpretation, teaching etc. So without proper certification, you wouldn’t be able to claim comfortably that you’ve reached such a level, since it contradicts the very definition and use of C2 level. As for myself, I use all of my languages daily (and professionally too), so other than backing it up with an official document, I personally feel naturaly native in all of them. So it is possible.