I'm a Registration Officer and I can tell you that it's low key very important to the majority of couples. I see double barrelling but it's unusual for mum to keep her name if she's English, and unmarried mums almost always go with dad's name in my experience. I'm pretty sure parents are more likely than ever to be unmarried and yet children still usually have father's surname by default. Unmarried mums often think that the name is what will give the child some rights in terms of parental responsibility but it doesn't, it means nothing. If she gives baby her surname, it can be changed after she marries, but if she gives baby father's surname and regrets it she will need a deed poll and potentially court order to change it. This is related to the archaic shaming of unmarried mothers that has changed relatively little since records began - upon parental marriage, a child is reregistered as if their parents were always married and their original birth registration becomes null and void, allowing the child considerably better social stature as a 'legitimate' child born in wedlock.
Women used to be considered property to the extent that our forenames weren't always used, and we were buried as 'daughter of -' or 'wife of -----'. Make no mistake, your name is your identity and it does not exist in a vacuum, our expectations have been shaped by centuries of men coming first. You're either your father's or your husband's. Names have interesting meanings as well, of course, eg a 'Baxter' came from bakers centuries ago etc but I see so many interesting women's names get lost. I write death certificates for women with multiple previous names because of her divorces, while a multiple-married man has no indicators of past wives on his certificate. No one gets married thinking they'll get divorced but it happens for an awful lot of us. There is shame attached to that for women in a way that there isn't for men. Don't tell me that we have the same attitudes to men and women, that a name's just a word and it doesn't matter.
When I got married I really wanted to combine our surnames and make a new name. My identity is important to me - I'd had my name all my life, I think changing it is a big deal. I wanted us to be our own little team with the children we'd have but I didn't want to double barrel as I felt a long name would be annoying. Husband disagreed but didn't pressure me to have his name, so I compromised and double barrelled. He kept his name at first but changed it after 3 or 4 months. We're getting divorced now but we're both keeping the double barrel. If either of us remarries we might lose the name we share with our child for the sake of our new family. If we'd changed it to a combined, made up name instead of double barrelling, that wouldn't be a problem, so I do wish we'd done that. But then we would have lost our two long-established family names for something without historical meaning, so there's no perfect answer.
Different cultures do it very differently and in many countries women don't change their names, but that's not to imply that any countries don't have male dominance woven into them. Unless we start afresh with a whole new naming system, I don't see it improving anytime soon in the UK.