A popular term in the UK is "continental Europe" or "the continent" for the mainland, whereas the islands are considered to not be "on the continent". Continents are basically a made up structure anyway, their definition is more by convention than geology.
Yeah - in this context it's a pretty poorly defined term... Particularly since the original comment could be read as either " anyone on the continent (that geographers call Europe) ..." or " anyone on 'the continent' (where you get continental breakfasts instead of full English breakfasts) ... "
I'd probably rather question whether the ".eu" tld means "Europe" (in which case there's a reasonable argument to be made that UK residents should be permitted to use it even post Brexit) or if it means "European Union" (which'd mean the UK voted to opt out of being able to use it).
From the perspective of the UK, "the continent" means quite explicitly "not the UK". Geographic reality not withstanding.
They feel rather strongly about this and other issues that might not seem entirely correct to people with a different perspective. I suppose that's what ultimately caused this situation in the first place.
A popular term in the UK is "continental Europe" or "the continent" for the mainland, whereas the islands are considered to not be "on the continent". Continents are basically a made up structure anyway, their definition is more by convention than geology.