No, it sounds like an affectation to me. There is a very small “English nationalist” movement, something of a reaction to nationalists in other constituent countries, who claim to identify as English rather than British, but I’ve not heard it the other way round.
From what I understand, the English nationalist movement is a pretty right-wing, anti-foreigner, “England for the English” thing, so I can see people not wanting to be associated with it. I can’t think of a good US analogy, but I guess it might be like a liberal from Atlanta saying “I’m not from ‘the South,’ I’m from America!” Obviously, they’re from the South, too, but they want to make it clear that they identify with the country as a whole, not with some particular part of it.
I think it’s more a case that English nationalism tends to be tied to right wing politics. The concept of Britishness seems to be more inclusive of all the nations of Great Britain (and much of Northern Ireland ) as well as the UK’s ethnic minorities.