You are right: here in the EU we often refer to ourselves as 'Europeans', thus implicitly excluding people from the non EU part of Europe. That's inacurate (and a bit disrespecful to non-UE Europeans, if you ask me). For that reason I try to always use the expressions 'EU' and 'EU citizen' instead of 'Europe' and 'European'.

Which reminds me that USA != America, also. It would be perfect we should not have to deduce from context when 'American' refers to anyone from the whole continent, from Canada to Chile, or just someone from the US. I guess it would be more practical to use 'US' and 'US citizen', for clarity...