Thank you, this is the answer I was looking for.
What makes the work unique to each miner?
With Stratum, each connection is assigned unique identifier that is inserted into the coinbase transaction. This prevents two connections from hashing the exact same headers for the block since the coinbase will never be the same on two different connections.
Since each connection has its own unique identifier, you couldn't route your hashes to different connections running at different difficulties, since the hash would only be valid on the connection assigned to that identifier. It's similar to how you can't take a block solving share and submit it to your own wallet, since changing the destination wallet of the coinbase would change the hash, meaning it no longer would meet difficulty.
Obviously, this all assumes the stratum implementation was done properly. AFAIK, no pool has ever been known to have screwed up this specific part of the stratum spec.