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I would assume there is enough data sent to fingerprint each miner in the same way browsers are fingerprinted. I would only be confident there's not if I was proxying my own data.
There is no fingerprint.
Unless you are using an aftermarket firmware that sends extra data to the pool that is not part of the stratum protocol,
and the pool is designed to accept this non-standard data,
(e.g. any aftermarket firmware that is closed source, who knows what they are sending and where they are sending it),
then there is no data that can 'fingerprint' the physical miner.
The standard stratum protocol that all miners must support has zero data that can identify an individual physical miner.
Hack firmware, and pools that support those hacks sending identifiable data, on the other hand ...
