Hi Michal,
Thanks for posting back.
Regarding your query, I searched all official documentation but didn't find anything that could affect the "Show Bar" feature. If you also can't view it on your side, please sign out all accounts in the Office applications, restart your device then check again. Also, please note that the sensitivity bar is another feature that can be found on the title bar of the app. You can view its feature name and its image in this article.
For "another issue with labels not getting synchronized", did you edit its name or other settings? Normally, it may take up to 24 hours to take effect.
For the last question, I also searched and tested on my side, but the conclusion and the result are opposite.
According to this article, the sensitivity labels are based on the primary Office account shown in the top right corner of the app. So, it should be changed when switching the account.
Based on my test, even if I signed out Account A, deactivated Office, then signed in Account B to activate Office, I could still see Account A's sensitivity labels.

Based on all of the above, to troubleshoot further, I would like to suggest you raise a support ticket via Microsoft 365 Admin Center> Support> New service request. That support team there will have the correct channel and resources to help you investigate more and find what exactly the reason has caused this situation.
Appreciate your understanding and patience.
Sincerely,
George | Microsoft Community Moderator