That's a bit different though. It's typical in Ireland and the UK to refer to mainland Europe as 'the continent', whether they consider themselves European or not. The likes of the Faroe Islands, Iceland, Cyprus, Malta, the Channel Islands, &c. wouldn't be considered part of 'the continent', though some might consider the Mediterranean islands to be 'continental'. 'Continental Europe' is not the same thing as 'the continent of Europe', but a subset of it that excludes its island parts.

This is not unlike the way that the 48 contiguous states of the US are considered 'the continental US'.

Frankly, I find the attitude of some English people towards the continent and Europe as a whole to be somewhat silly.