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When a collator passes a PoV to a validator, how does the validator know that the node is a valid collator for a given parachain, and that they had permission to author a block in that slot?

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The relay chain validator can not know this and doesn't know this. It will accept any PoV and it should fail at validation if the author isn't the correct one. This is the job of the validation code to ensure that only eligible author(s) are producing blocks.

In the future there will be some changes on the relay chain side to have some lightweight check. This would be used by the validators when a collator opens a connection to ensure that this collator is eligible to build the PoV. This check could for example be the verifying of the signature in the header. Putting this into some extra check before downloading the actual PoV reduces the bandwidth requirements as you only need to send the header plus a proof to verify the signature.

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