It is funny how in the US we refer to "Europe" like it's a country when it is, of course, many countries. I think it's because many of us think how something is categorically different there without thinking of the nuances. For example, health insurance is different in all of Europe than in the US, but the differences between countries are not thought of as often here. I'm sure it's actually rather offensive now that I think of it to omit the national identities.
Do Europeans think of "North America" or "America" to mean the US, Canada, and Mexico or just the US?
I think it is hard for Americans to realise the fast cultural differences in Europe. The difference between US states, or even between the USA and Canada are much smaller then the differences between European countries.
Normally I do not really mind when a person says European for something that is absolutely not common in Europe, just in one or two countries. The poster who said European instead of Brittish when it came to words went a bit too far in equaling one country with the cotinent in my opinion.
When people say America, they usually mean the US. The difference here is off course is that USians themselves refer to the US as Amerika.
If you say North America, most people I believe think of the US and Canada (In my experience, not many people know that Mexico is part of North America)
After having heard from some South Americans that they do not like that people associate the word America and Americans with the USA and USians, I have started trying to use the words US/USA/USian
Slightly different subject, but I thought it was also here that a person refered to football as European football. It just made me wonder how the South American, African, and (for as far as I know) Asian countries feel about that.