My company required an vaccine status acknowledgement - said it was only to be shared with HR, and only used to determine return-to-work plans based on location. It was explicitly stated that your manager would NOT have access to the information.
I told my boss then that I didn't expect this information to be handled appropriately or in the manner we were being told - but I was on the way out for vacation and trying to not make his life miserable, so I checked the box.
Three months later, we've introduced our new 2022 return to work plan (that is largely in line with the OSHA ETS). Low an behold, the unvaccinated are required to have weekly testing, and of course your manager needs to know if you're required to test and if you have a positive result. Oh, and of course we have to share that information with the third party testing company as well.
If they want the data, and aren't asking for it through a truly anonymized source, they're likely going to use it for something, and you can't take that pee back out of the pool once it's there.