You misunderstood the whole situation. The signatures are all fine. Google Play Store is trying to override an app installed from F-Droid. If the two stores had the same signature, the play store would be able to do this which would go completely counter to the user's choice (they installed from F-Droid for a reason). It's a good thing the signatures don't match, there's nothing suspicious about it.
It used to be that the play store just wouldn't show updates to apps that it wasn't actually able to update. They broke this behaviour.