Plenty of people are British - Scottish dad, half English, half Welsh mum, or whatever.
English has a sort of odd stigma though - a bit like being middle aged, middle class and white - too interestingly privileged and stuffy.
No different from all the Americans you meet in Ireland in tourist areas telling long suffering tourist guides or shopkeepers how they're Irish because they have one Irish great, great grandparent - completely ignoring the other 15 great, great grandparents who were all from boring, stuffy places like England, Germany and the Netherlands...