Here's a quick demo using Live Share and recording the windows side-by-side.

I just tested it and yes, I was able to open the Live Share link and have two sessions side-by-side. However, the light/dark theme wants to synchronize between the two windows. I had to use the "Workbench: Color Theme" setting to change just one window.

You could try Live Share to synchronize two instances of VS Code and record them simultaneously.