You can actually sign the F-Droid app yourself, if you use reproducible builds.

There's reasonable odds the signatures still won't match though, because Google requires App Bundles now, and then they build and sign the APK, rather than allowing the developer to build and sign their own APK.

Technically you can use the same key (see "Best Practices" of this page), but it's kind of shady, and requires giving your private key to Google.