Hi Stefan,
Here is the update:
Generally, after enabling sensitivity labels for SharePoint sites, admins and end users can select sensitivity labels when they create modern team sites and communication sites like below:

When the label is applied, and users browse to the site, they see the name of the label and applied policies like below. When the mouse is placed on the label, it only shows "javascript" in the lower left corner.

For your requirement to make these sensitivity labels invisible, I researched a lot and tested on my side, however, I couldn't find any settings or methods to change it. So I am afraid we can’t make these sensitivity labels invisible at this moment. I understand that it would be much convenient for you.
Given this situation, we recommend that you submit requirement for this feature in this SharePoint UserVoice and many members who has the same requirement with you will vote it in the future.
Appreciate your understanding.
Best regards,
Jazlyn