To be more consistent with Post Editor, we should consider automatically switching the sidebar between tabs based on context. We already have a feature in place that automatically switches to Block tab when opening the settings sidebar while a block is selected.

In the post editor, when we select a block (while nothing is selected), then the sidebar automatically switches from Page to Block. And vice-versa: when we remove our selection then it switches from Block to Page. I'm not sure if this is intentional though. So it might be a bug in the post editor instead of a feature. cc @mtias @jameskoster

For reference, here is how the post editor works: (Note: It doesn't switch to Block tab if we just keep changing our selection between blocks. But it does if we have nothing selected then we select something)

And here is how the site editor works: