They certainly know the address you used to make the deposit and some other information they can get from browser fingerprints like your ip address, os, browser.. Most of these information can be obfuscated by using a private browser which is highly recommended when dealing with mixers.

Most if not all mixers aren't licensed so they aren't under any obligation to disclose their customers' data to any third party. No one would be interested in using their services if they do so, anyway.
Besides, most of them claim they don't store their customers' activity logs since they clear them as soon as the session ends.

I doubt any reputable mixer would do something like this but technically it's doable. All they need to do is to seize the deposited money without sending the mixed coins.