Yes they do A message appears saying "(Name) sent you a gift" something like that anyway and they get prompted to send one back
I was just sent a gift, I have not leveled up and there was a new exclamation next to the hanger, and I had a new dress that someone gifted me, but it did not tell me from whom. So I can't return the favor
Yeah sometimes the gifting notification can glitch but rest assured this is very rare. If need by change your character's greeting, which you can do in the wardrobe by clicking on the bubble left of their head, saying thank you for the dress but don't know who sent it. That's what I do when this happened to me
Comments
I was just sent a gift, I have not leveled up and there was a new exclamation next to the hanger, and I had a new dress that someone gifted me, but it did not tell me from whom. So I can't return the favor