Interestingly the reason I know this is because I’ve read the Poetic Edda.
It was traditional in many Northern European cultures to gift a child a small amount of money…a
tand-fé, or Tooth-Fee…when they lost their first tooth, as a small celebration of a life milestone achieved. Also because the Norse thought that a child’s tooth worn into battle as a talisman would bring luck.
(If nothing else, I suppose that wearing one of your children’s milk-teeth as a reminder of what you had waiting for you at home would be good motivation to, y’know, not fucking die. Sort of like taping a picture of your kids into your helmet, like modern soldiers do sometimes.)
ANYway, moving forward in time, there was a lot of superstition around teeth, and they were often either buried or burned to prevent them being used to work Bad Mojo on kids. It was found that it was easier to get kids to give up their teeth if you paid them off, probably by some dude in a tavern going “How the fuck do I get my kid’s milk teeth away from him once he loses them so that I can burn them to prevent a witch from cursing him” and some dude of Norse descent going “Just pay the kid off we’ve done that in my family for like, forever, works a charm.”
The story of a fairy creeping in to take them and leaving money was a fairly modern addition, probably intended simply to be a fun story to tell kids and to make the standard bribe-kids-to-part-with-teeth-with-money practice a little more fun.
Also, never ever actually give any fey any part of your body that is a VERY BAD idea.