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Tooth doctor doesn't have the same ring as eye doctor
(lemmy.world)
A "Showerthought" is a simple term used to describe the thoughts that pop into your head while you're doing everyday things like taking a shower, driving, or just daydreaming. The best ones are thoughts that many people can relate to and they find something funny or interesting in regular stuff.
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.