I think the simplest answer is that no, you can't at least easily run another instance of Steam via the Flatpak. The two instances would need to hit separate configuration folders and so on. I think what might be possible is creating basically a full-system container with a separate /home and installing Steam there. You would then need to pass the X11/Wayland connection and other such stuff into that container so you can get graphics working. I think Distrobox might be able to do it, or you can do it the hard way directly with Podman/Docker or systemd-nspawn, LXC etc.

It would be a very roundabout way, but I guess technically you could run the native Steam client, and at the same time, run the Windows Steam client in Wine...