Things like this are not uncommon.
All throughout their school years, this can happen.
Sure, send a thank you note. That is proper.
And to acknowledge it, the gift.
Sure, you can get a gift in return for the gift he gave your daughter.
That is up to you.
As you said, your daughter and he, are friends in school.
Gifts, if given to particular friends, ideally, yes, you give it to the other kids, discreetly. Not in front of the other kids.
But again, this cannot always be, done.
If anything, I say that this is between the adults, and having the child give it to their Mommy and that it is "private."
But again, some kids, because they are so young, have big mouths and they don't mean to. They are just excited to see a surprise given to them or in their cubby and may exclaim "look! I got a present/surprise/etc. !"
Or... you just give it DIRECTLY, to the Mom. To then take home. For her child.
If gifts are given at school, because that is the only place to drop it off at and give it to the child, then, I personally put the gift in a generic plain ol' bag.... so that it is not attention getting. Then, I tell the Teacher and/or give it to her, to give to the student, just before school gets out.
Teachers are used to this.
They know how.