I think there is no such specific shortkey - but it might not be needed if you consider the range of options available without trying to get exactly what you know on other desktops.
Especially considering the workflow:
I think Pantheon workspaces are rather intelligent in that there is no new empty workspace and any fullscreen window can get its own workspace (Super-F); use the shortkey that shows workspaces (Super-S, Super-DownArrow, those can be customized) and there you can drop windows of the same application onto the empty desktop (the + button) to have a separate workspace for them.
You achieve the main purpose if you use the multi-window application (and only that) in its own workspace and then switch windows there with Alt-Tab. If you need more than one multi-window application, use separate workspaces for each app, and switch between apps by switching workspaces (Super--Tab).
The quickest way for me is this: starting first window of my application, then Super-F to get a new workspace, and any new window of that app will stay in that particular workspace; if I need a new multi-window app, start the app, Super-F again to get a separate workspace for that and then open the second window etc. If the fullscreen window is taken out of fullscreen (again Super-F) it might go to the previous workspace but it can be easily moved as desired either by going to the workspace switcher (Super-S, Super,DownArrow) or by using the shortkeys that move windows between workspaces.
Is it better?
This may seem complicated at first, but in fact it may prove easier than what you ask: if you had the shortkey that you mention and you used more than one multi-window applications (let's say Libreoffice Writer and Evince), that shortkey would only work for the selected application; to go from LOWriter to Evince you would either use the mouse or set separate workspaces for the different apps in the first place, in which case your ubuntu shortkey is useless.
I think this is just a matter of using workspaces or not.