Tooth doctor doesn't have the same ring as eye doctor
Saying eye doctor in a conversation feels normal. Saying tooth doctor feels like listening to someone who had a bad disco elysium roll and they're punching themselves in the face to get the words out.
The one that always bothered me is Hals-Nasen-Ohren-Arzt. Why is it Nasen? We only have one. It should either be Hälse-Nasen-Ohren Arzt or Hals-Nase-Ohren-Arzt.
I assume it's for the same reason you say Nasendusche, Nasenrücken or Nasenbluten but Halsschmerzen or Halstuch (though I'm not sure what's the exact grammatical reason). It has nothing to do with the number of noses. I think it would be Augenarzt and not Augearzt even if we had only one eye.
But that's because you learned english and know it's called a vet.
Direct word-for-word translations are often supposed to be weird because that's just not how it works.