Nazi is not a political ideology, and never has been. It’s an insult and always has been (*). The correct term would have been National Socialist, or a member of the NSDAP (Nationalsozialistische Deutsche Arbeiterpartei). When they were in power, it was expanded to include all Germans, after WWII it shrank a bit to only those that were actively a member of the previous government or supported them. So, Nazi was an insulting abbreviation at the time, and has become just a plain insult since.
As an insult, the only defining characteristic that it has is that someone doesn’t like the person or group being insulted. Feminazis, grammar nazis, whatever.
Ignoring your specific example, and generalizing to when can you say that someone is a member of a particular political ideology, regardless of whether they profess to the ideology or not. The answer is simple: zero percent. After all, you are being proscriptive not descriptive, labeling someone else, who cares whether they or anyone else for that matter agrees with you or not.
If you want to be accurate, you’ll describe characteristics not apply political labels. If someone professes to an ideology or party or to an opposing ideology or party, you can describe the characteristics in relation to that ideology or party.
Otherwise your are just being insulting, which, hey, is ok, there’s a whole world of people to insult, most of them deserve it.