Socially yes it can hurt a little. Though I'm such an introvert that living on my own with housemates and at home, there isn't much difference. I guess my social needs are a lot lower than most others.
As for the finances, it's huge. I've only worked for 7 months and have accumulated $45K as a per diem worker and grad student. If I was working full time, it could easily be $60-70K in savings right now, a huge amount. Right now I'm near $65K hoping to hit 80-90K end of this year and the dream is $200K coming out of grad school (later 20s).
Technically if I stayed at home and kept my expenses low by freeloading off parents (who are okay with me not paying them anything because they have a pretty nice retirement) then I could retire by 36-38, assuming no big events like marriage, kids, etc. If I delayed all those things, I could retire early, return to working part time, per diem and raise a family and spend time with kids or other activities. Am I willing to sacrifice that much for FIRE? Probably not. We live life once and I'd like to enjoy some aspects of it, which means I'm not going to stay at home forever. In fact I'll probably move out this summer. Living at home for 1 year working full time really helps build a nice cushion and platform. But once you're on your own, it's a cold world and your savings rate dramatically decreases. I would say my savings rate is around 85-90% right now.
I do feel I lose some independence but the savings is huge. I often argue in my head, "move out, get independence" or "stay home and accumulate $15000 more?" I want to move out and have that pressure but the same time I really like the money. But in the end the need to move out is kind of overpowering so I tell myself to hit my financial goal first before moving out. So $75K then I'll move out. Hopefully I don't keep changing that goal!
It depends on your goals whether you want to stay at home or not. If you want to FIRE, then yes staying at home for a bit is a good idea. As you build up your stash, it also teaches you the value of saving and when you move out, you'll be less likely to spend as you have developed those habits as opposed to moving out first thing and not having those habits. If social life is more important, than staying home might not be worth it. It depends on you as a person, how much you need to "live", and what are your goals.
There's definitely stigma but it's becoming more and more acceptable I find to stay at home with your parents for a bit. I guess 26 is sort of the limit but some go out further, even to their 30s. I personally think people should do what they want and what they need to do to survive and reach their goals, even if it might be stigmatizing, it's better than drowning in debt. On these forums especially, it's more acceptable as we all try to build our stashes.