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I've no teeth and my tongue won't lick,

My collar's got no neck, the trick

Is; to use someone else's me.

You'll surely see what they would see.

And if I'm on the other, wish!

My essence is a fish.

'Ha. Your essence is a fish? Does that mean if you're not flipping out, you just flop?'
'I said fish, not shoe.'
'Oh, so it means you smell really bad?'
'Essence. Essence. Not incense or whatever nonsense you're thinking.'
'So is the answer a smelly shoe or not?'

Riddle Answer A Shoe

Riddle Explanation
to use someone else's me. You'll surely see what they would see - to put yourself in someone else's shoes.

and if I'm on the other, wish! - the shoe is on the other foot.

my essence is a fish - a soul and a sole are homophones.