Yes, I can replicate the issue. If this was intentional, then I think it's another unnecessary restriction regarding the now playing panel layout.
(A while ago I reported some buggy issues related to the panel, but don't see much differences since then: http://getmusicbee.com/forum/index.php?topic=11333.0
Anyway a workaround would be:
- Choose another tab.
- Open now playing config pane, turn off now playing list, and tick "stack panels using tabs".
- Click on now playing node in left navigator, then you will see the tabs remain on the sidebar even after restart.