No, I have no problems in saving it. It simply saves as "unnamed skin".
After restarting the Launcher, it seems the name field is active and I can type in a name. But if I go away from the Skins screen, when I'm back it will show again "unnamed skin".
I'm having the same issue.
Using MacOS
Launcher version:
2.2.2310
Wednesday, March 10, 2021, 1:49:16 PM
6cd58158ae776414beb2bcb966139b5af724e5ef
Apologies, it seems my son (who is very young) managed to raise this ticket on his own using my account.