I was born in Wales, to parents born in Wales, went to school in Wales until age 13, so learned Welsh in school, but don't speak it fluently. I have lived and worked as an adult in both England and Wales, raised my first child in Wales, and currently live in England with my English partner, and my son who was born in England.
My nationality is British as per my passport, and I mostly think of myself as British, and I honestly don't think there are significant cultural differences between Wales and England. (I think there is just as much difference within England and within Wales is there is between the two countries. )
I know many Welsh people who fervently think there are though, including my father, who still lives in Wales and strongly identifies as Welsh, as do most of the people in his community. Almost none of these people speak Welsh (the majority of Welsh people don't speak the language). And many of them, like my father, are descendants the huge number of immigrants to the South East Wales coalfield in the late 19th/early 20th centuries, so have no Brythonic celts or welsh speakers in their ancestry. They can't possibly share an 'ethnicity' with welsh speakers in rural North Wales who may have had ancestors living in that area for thousands of years. What they do share is a strong identification with a place and a community.
If OP's husband's welsh ancestry is important to him, and an important part of his identity, its not up to her to tell him he's wrong.