Very interesting. No, I do not think so. C2 level entails a firmness not only on points of grammar, but most importantly requires a rather in-depth engagement with cultural points of a language. At C2 one is pushing for an acquaintance with proverbs and subtle linguistic turns that go far beyond the knowledge of many native speakers.
On grammar alone, as has already been noted, many native speakers of English (my language) regularly use non-standard (you may say 'incorrect'-although I do not) forms. For example in London you will regularly hear 'I ain't got nothing' which is a 'non-standard' version of 'I do not have anything'. A person taking the C2 exam would lose marks for saying this: nevertheless it is absolutely 'standard' in many parts of the English speaking world. Additionally many of the finer points of English grammar are not taught to the majority of children in England, with the consequence that they make 'mistakes' in points of English grammar that even a B1 student would have under control.
Culturally it is absolutely clear that many C2 speakers of a foreign language have a far firmer control on cultural and historical points and are finer tuned to linguistic 'register' and stylistic points than many native speakers. At C2 one might engage with certain poetic flourishes or strategies in argument construction, skills that many native speakers do not have.
At C2 level one speaks the target language (more or less) as an EDUCATED speaker of that language might: but remember that educated people tend, numerically speaking, to speak a very NON-STANDARD (often slightly archaic) version of their language.
Native speakers, however, have the advantage of being linguistically comfortable in all (ordinary) life-circumstances, e.g. negotiating with a car-mechanic, that would still cause a C2 speaker difficulties. Lastly, many C2 speakers often still use very conspicuously odd-sounding turns of phrase that no native speaker of the language would ever use.