Does Steam Deck not have rpm-ostree (or an arch equivalent since RPM is fedora-specific)?

Steam Deck has a custom solution involving an A/B partition scheme of immutable btrfs filesystems and overlayfs for layering changes on top of that.

Also, what about distrobox?

If there's a way to install containerization software with Flatpak, maybe. Docker isn't available out of the box, though.

I haven't really tried to do anything package manager-related on my Deck, so I'm going on what I know from Bazzite, but there are several ways to install non-flatpak software on it. In fact, I even installed yay on an Arch distrobox, and I can install things from the AUR (as well as the official repositories).

You can use pacman, but it's volatile and requires making intentional changes to restore its functionality.

The first option is to disable the read-only flag on the root filesystem, then set pacman back up so it can pull packages. Whenever the root filesystem image is updated, you'll lose the changes, though.

The second option is to add an overlayfs to persist the changes in a different partition or inside a disk image on the writable storage. There was a tool called "rwfus" that did this, and it worked well enough if you were careful. If you ended up upgrading a package that came installed on the base image, though, it would end up breaking the install when the next update came around.

With all the caveats, when Valve made /nix available as a persistent overlay a couple of years ago, I just bit the bullet and learned how to use Nix to install packages with nix-env -i.