Why was this, does anyone know? i.e. why weren't they William/Harry Windsor-Mountbatten?
I understand it was their father's main title at the time, but aristocratic families have a surname as well as a title. e.g. in downton abbey the Earl's actual name was Robert Crawley, not Grantham (I know that's not real!), the current duke of Westminster is Hugh Grosvenor not Hugh Westminster, etc.
Anyway OP, I suppose the thing is with Eton is lots of the pupils will be not-ordinary. So even though George is unique in eventually being king of the country it's in, there will probably be some other pupils there (particularly those from abroad) whose parents are either significantly richer or more powerful in terms of being leaders of state or heads of multi billion organisations, etc., as well as other royals.
Same to a lesser extent with their primary schools. I doubt any of their classmates are going for an unaccompanied kickabout at the local park or knocking on their mates door to ask if they can come out to play, with or without Louis and Charlotte. They probably can have playdates, just the families will be vetted first and security might hang around outside. But, again, it's possible, even likely they won't be the only kids in that position.