Firstly I want to say that it’s great to see this feature introduced. It’s very useful. Great work from the contributors!
I think all of the points made in the above discussion are valid. I can see how “profile description” would be appropriate but also possibly mistaken for a bio. I agree that the privacy part is explained in the dialog, though only after selecting the menu item. And clearly the German localization can be refined.
The other two menu items (“Mute Notifications” and “Delete”) include a verb, so maybe it would make sense to do the same for the label, for example “Add (Private) Label” if the label hasn’t been set or “Edit (Private) Label” if the label has been set?
Aside from the wording of the menu item, I’d note that it might be helpful to visually distinguish the private label from the profile name more in the profile switcher. The screenshots in Asiel’s pull request show the private label and the profile name in distinctly different colors, but on my device, at least with dark mode enabled, they show up in the same color and the font size is only slightly different.
I also think that some people could find it more helpful to emphasize the private label instead of the profile name (display the private label in larger font and the profile name in smaller font) but I’m sure people will have different opinions about that.