I think it's just an odd mind-set we have. I have an odd dual attitude - I'm very pro-Europe (and think we should switch to the Euro, join Schengen etc) yet I still think of Britain as geographically separate from mainland Europe. Although logically I can see that in effect we are an offshore island of Europe. I don't think I'm that unusual in that...
Perhaps it's generational. 'when I were a lass' we definitely were NOT part of Europe and we tend to retain impressions that we grow up with, even though the world changes.
In any case all this is not very relevant to the fact that the accepted names for the two main variants of English are American English and British English.
According to the BBC Education site
It is estimated that over 95% of the British population are monolingual English speakers. There are various minority Celtic languages, and speakers of these are invariably bilingual English speakers. In Scotland 1.4% speak Scottish Gaelic as well as English; in Northern Ireland 6.6% of the population are bilingual in Irish Gaelic and English; in Wales, 21% also speak Welsh. Welsh is the only Celtic language that enjoys official status.
In Scotland, Northern Ireland and some border areas of England, Scots is a distinct minority language although at times it overlaps with Scottish English.
There are also large numbers of community languages, brought into the country and sustained by recent immigrant communities, which account for more than 5.5% of the population. The largest group (spoken by 2.7% of the total UK population) are South Asian languages such as Bengali, Punjabi, Hindi and Gujarati. Other community languages include Cantonese, Italian, Polish, Greek and Turkish. 45% of the total ethnic minority population lives in London, but community languages are spoken throughout the United Kingdom.
see http://www.bbc.co.uk/languages/european_lang
*Moving to Linguistics section for those who wish to continue the debate*