The seller will know your name and your bank during the trade. All that matters is if you're ok or not to let someone knows your real name.
Zelle has no way to know what you're doing (buying bitcoins) unless of course, you add a note saying something like " for my bitcoin purchase"

Then depending on the P2P website you use, the same information will be stored on their server. Let's say I'm an agency tracking you, I will just need to contact the website to ask everything I need
This is an open door.

Unless you use a decentralized exchange like Bisq where everything is stored on your own computer and not an external server. This is not an open door but if I know the seller, I can convince him to snitch your detail