StackOverflow says that it can be done by editing xrdp.ini:
https://askubuntu.com/questions/133343/how-do-i-set-up-xrdp-session-that-reuses-an-existing-session#360835

Is there a specific reason for choosing RDP?
VNC is the most common protocol in Linux. And RustDesk is also a good, more advanced alternative.

which generative model did you use to generate this post?

your best bet is probably to take a look at Lutris. Both client and website. And then go from there.
Think about how multiple instances of the website should interact with each other. And think about how the client would work with such system.

Also, aren’t there a few open source Steam clients? See if there is anything useful in those projects as well.

ah yes, that, wormhole.app, that is closed source. (but if I am reading this correctly, some early iteration of it is open source. https://github.com/saljam/webwormhole )
Magic wormhole is a different thing.

can you link to the post that claims the protocol is not open? I’m interested in looking into that
anyway, source for the magic wormhole can be found here: https://github.com/magic-wormhole/magic-wormhole which also links to both the Mailbox code and the Transit Relay code.