That is the behaviour that’s built for when an upgrade through a “classic” package manager (e.g. apt, dnf) updates Firefox while it’s still running. The only way I can think of that you’d get that with a snap is if you’re intentionally bypassing the confinement (e.g. by running /snap/firefox/current/usr/lib/firefox/firefox directly, which can also massively mess with other things since Firefox won’t be running in the core22 environment it expects).

If you’re using the snap as expected (e.g. opening the .desktop file in /var/lib/snapd/desktop/applications/, running /snap/bin/firefox or running snap run firefox), snapd won’t replace /snap/firefox/current until you no longer have any processes from that snap running. Instead you’ll get a desktop notification to close and restart Firefox to update it, and two weeks to either do so or to run snap refresh --hold firefox to prevent the update (or something like snap refresh --hold=6w firefox to hold the refresh for 6 weeks). Depending on what graphical updater you have, you may also have the ability to hold the update through that updater.

Are you sure you’re running the Firefox snap? Because that sounds pretty much precisely like the expected behaviour if someone had gone to lengths to avoid using the snap.