The executable in /Applications is 32-bit, but it's just a launcher for the real Steam app in ~/Library/Application Support/Steam/Steam.AppBundle which is 64-bit.

Ahh thank you! I knew I was missing something... I'd get the alert message, then the Steam app would seemingly close and re-launch. I wonder why it is architected like that.

I guess that it's like that in order to be able to update the executable for non-privileged users (who can't write to /Applications), which could be e.g. kids' user accounts in the family computer.