I am an Englishman, so I'm not American.

No problem here, or did I misunderstand your question?

Well, you can also say, "I am American," or "I am a Briton," or "I am a Frenchman." However I think their may be a trend you are picking up on that has to do with language. The "new world" countries have names that developed in the time of modern languages, so different forms of the word fit more neatly into our grammar. It's just a convention of the language. From my limited experience with french, I think a french-speaking Canadian would be more likely to say, "je suis canadien," than, "je suis un canadien."

Actually, in both cases you're using an adjective. What you're calling a noun is in fact a substantive adjective, which is an adjective being used as a noun with the usually-accompanying noun (in this case man/woman/person) omitted.

English convention only allows us to do this with a few words, so American works but we have to say 'Frenchman'. However, we can still use it appositively, eg. "He is French. "