These links might help Search results for 'chat app' - MIT App Inventor Community may give you ideas.
What have you tried?
This might be as simple as changing the Tag
Just save your text to a different tag ... this shows chat is one room. Use a different Tag to store chat#2, chat#3 and provide a way to move among the rooms (put the chat room names in a List and use a ListPicker or ListViewer to go to the room of your choice. All this should be possible with a single Screen. What you will probably need is a Label component for each 'room'.
You will need a set of CloudDB blocks for all the Cloud controls or a way to use a variable for the Tag . Either method should work. The important part is that you need a 'different' set of controls to maintain each 'room'. However, with a little ingenuity and using a variable to select/store the chat room name, you might be able to use a single set of CloudDB commponents., only changing the chat Tag as appropriate.
Why not experiment. There is not tutorial to do this. When you get stuck, share your blocks and someone should provide specific advice.
Regards,
Steve