Please verify that it works correctly in your project. If so, you can compare it with your own setup.
The two cylinders on the left do a bubble conversation. Click on the cylinder closest to the middle to start the conversation.
The cylinder on the right does a regular screen space conversation. Click on it to start the conversation.
Some notes on how I set it up:
There are 4 actors with Dialogue Actor GameObjects: Regular Player, Regular NPC, Bubble Player, Bubble NPC. The Bubble actors' Dialogue Actor components are configured to use overhead bubble panels.
The Dialogue Manager's Subtitle Settings > Continue Button is set to Always.
The Bubble NPC has an Override Display Settings component to test overriding display settings. Continue Button is set to Never.
The Bubble NPC also points to a Standard Dialogue UI (colored blue) so it doesn't have to try to share (and conflict) with the Dialogue Manager's Standard Dialogue UI (colored default gray).
And I'm using the new Input System package, which could be where the problem lies.
After opening your "Simultaneous Conversations Example", I swap the input type to the new Input System in the EventSystem, then I change the UseKey in the Selector script so I can initiate conversations.
The problem with current subtitle line disappearing when another conversation starts still exists.
After finishing the current conversation, the cyan "Dialogue UI For Bubble" under "Bubble NPC/Canvas/" kicks in, which I assume this is a dummy UI and should not be used.
This can be prevented by adding a "Override Display Settings" to regular NPC, but I still don't want the current line to disappear.
Also, what I had in mind for the Bubble conversation is done between two NPCs. In the model you provided, the Bubble conversation also goes away after choosing a choice in Regular conversation. However, this won't be a problem for me, as I intend on letting the bubble conversation happen only among NPCs.