@ParentsandParents - that's quite similar to us. My DP had the baby because she was older, but also because I have a history of pregnancy loss. I hope you manage to get your health issues sorted!
As to how I've found it ... TBH, it was a shock. We had lots of microaggressions/ bits of homophobic treatment that added up. I do think there was a perception we were doing it 'wrong'. A weird thing I found was that, although everyone says it would be wonderful if more couples shared the care for a newborn, in practice, there is a lot of hostility or confusion. One time DP took a well-earned nap and the HV turned up to weigh DD. It was explicitly a visit for DD not DP (DD was under extra observation) and we'd checked this was so. But still, the HV was very unhappy that it was me not DP and clearly found it totally inappropriate. If she'd said she wanted to see DP to discuss breastfeeding or something that would have been fine, but we'd been specifically told she didn't!
That stuff gets a bit better once you're past the newborn stage but it doesn't actually go away. DD is 4.5 and all social groups are still basically gender split. I have got better at initiating conversation with the dads but it is always about making an effort and knowing you may get blanked/hostile responses.
The other thing DP and I both notice is that, long beyond the point where you stop talking about BF or other birth-mum-related stuff, social groups are uneasy with two mums. You mess up the dynamic by being a couple when everyone else is just on their own, or, if it's couples socialising, people don't know which of you to talk to. I find it really weird and it's clearly just social conditioning, but it is isolating.
I used to love MN, and I was on here for years before DD, but sadly I have to say it's either got a lot more homophobic, or it was always that way and I've just noticed. You will get a lot of threads where people take great pleasure in implying you're not a mum, or you are not really your child's parent. It's depressing but it's the case. There is still a lot of good support and I'm sure the majority of posters are not homophobic, but it has been a real eye-opener.
I suppose the thing that gets me is the casualness of it all - eg., DP and I recently did some legal paperwork and one of the questions was who lived in the house. We said our daughter. He wanted to know who DD's biological parents were - it's totally irrelevant! Or, recently, when DD started at school, we found I was getting emails and not DP, and when DP said could they pop her on the mailing list, the reply was 'we usually contact mum' (which is, of course, sexist as well, but it's hurtful to be constantly treated as if 'mum' is someone who isn't you).
DP and I have tried to socialise with other groups of same-sex parents, although covid has made it hard. I really don't think the issues go away after babyhood, though. I wish they did. But I think what actually happens is you get a tougher skin about people insisting you can't possibly be mum, or your partner must be a man, or whatever. Sorry!