The desktop app on Linux can now share audio with your screen on Wayland. It has its own PipeWire audio capture, so the audio option no longer depends on what the browser engine supports.
When you share, the stream settings menu has an "Audio sources" picker, Vesktop style. You can share the whole system's audio, pick specific apps to include, or exclude apps from system audio. #2431 reworked that picker into its current form.
This applies to the desktop app only. In a browser on Linux, screen share audio still depends on the browser's own support.
If you are still having issues with this, however, please open up bug reports!
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Comment by Hampus