Is it correct to call myself a "European Portuguese native speaker"?

Yes, I believe that is the normal way of referring to that variety of Portuguese: European Portuguese (as opposed to Brazilian Portuguese or African Portuguese). If you think 'European Portuguese speaker' is a bit of a mouthful, you can always say 'a speaker of European Portuguese'.

I believe that, if you are from Portugal, you can just say "Portuguese". Brazil was colonized by Portugal so that's why we speak Portuguese in Brazil. I'm Brazilian and I always emphasize that I speak "Brazilian Portuguese" because, even though we (Brazilians and Portuguese) can understand each other really well, there are subtle differences in both languages, i.e., they are not significant differences, it is practically the exact same language, we do have different slangs and we phrase some sentences in a different way with a different pronunciation.

I guess it is more of a technical matter of who came first that should hold the right to state the name of the language without further specification of the region, as a Brazilian I've inherited the language and created a variation of it, hence, I speak "Brazilian Portuguese" and people from Portugal speak "Portuguese".