Is it possible to arrange multiple attribute tables in tabs in QGIS?

You put the attribute table into a dock widget in settings => options => data sources. Docks can easily be stacked as tabs.

There is still a manual process to it to stack the two tables. As far as I know it's not possible to do this automatically (yet).

In QGIS 3.14, they will open as tabs by default.

If the option "Open new attribute tables as dock windows" is active, new attribute tables will be open as tabs on top of other existing attribute tables.