I can confirm the issue. I tested on a Pi 4 with a Raspberry original 7" touchscreen running Volumio 2.779 and a second Volumio system in a virtual machine (x86).
Other than on Raspberry systems Volumio for x86 does not need a display plugin but a similar function already built in. Volumio x86 shows the same behaviour you described so it is not specific to the Touch Display plugin for Raspberries. In principle the plugin as well as the built in display function on x86 systems just run a local web browser in kiosk mode and display the same content as is shown in a browser on another device.
Unfortunately I don’t know much about how the multiroom option works and what happens server side when switching between rooms. As I can’t see a related error in the logs I have no idea how to resolve the problem ATM, sorry.
The same or at least similar issued has also been reported here, here and here.
For the time being you can also reload website hitting F5 if you can connect a physical keyboard to your device instead of restarting the plugin from another device.