On Ubuntu 20.04 using Vino, I was able to connect multiple VNC clients simultaneously to my server. With the move to gnome-remote-desktop and using Wayland, trying to connect to my server with a VNC client fails if there is already another VNC client connected from a different machine. This makes is impossible to access my server graphically if I've left an active VNC connection on another device and can't access that device now.
As the bare minimum, if a new VNC connection would disconnect the older ongoing connection, it's still better than the current scenario.
I found an insecure solution to the problem. The typical solution proposed on the internet for this is to set a blank password for the login keyring using the "Passwords and keys" app. This, however, is highly inscure as all the passwords on the device would be stored in plaintext.
What I decided to do instead was to create a new keyring in the "Passwords and keys" app with no password and set it as the default keyring. Then I proceeded to delete the VNC password from the login keyring. Then I rebooted the computer to ensure that the new keyring is the default one. After the reboot, I reentered the VNC password in the screen sharing settings. This would store the VNC password in the new insecure keyring. Then back to "Passwords and keys" app and set the login keyring to default again. Rebooted again and now the VNC password stayed saved and the default keyring was also back to the login keyring, ensuring all foture passwords saved on the device will be encrypted.
So in essence, I just reduced the insecurity of saving all the passwords in plaintext to storing just the VNC password in plaintext.