The game BattleCON: Online (792570) needed an update even though the Flatpak version of Steam just updated it. I am curious what is happening here.

Are you sharing a Steam library between unsandboxed Steam and Flatpak Steam?

If you are, one reason for games to need an "update" when switching between unsandboxed Steam and Flatpak Steam is that they are likely to have different versions of the graphics driver stack: unsandboxed Steam uses your host system's graphics drivers and libraries, but Flatpak Steam uses libraries from the Flatpak runtime and graphics drivers from Flatpak runtime extensions, and in general they will not be identical versions. This can result in different versions of the shader cache being downloaded: the shader cache is specific to one version of the graphics stack, and every time you swap between ways to run Steam, it is behaving as though you had upgraded or downgraded your system.

Or, if you are using separate Steam libraries for unsandboxed Steam and Flatpak Steam, they will behave independently, and will need updating independently (as though you were running Steam on two separate computers).