I needed to install ~tomcat/.vaadin/proKey
Ah, that was so obvious. This did not occur into my mind, since normally you do not need to. I was assuming you did production build WAR which you deployed to Tomcat. That is what one normally does. MPR (and other licensed software of ours) does check the license only in development mode, not in production mode.
Here is the chapter in documentation about setting up production mode:
https://vaadin.com/docs/v13/mpr/configuration/production-mode.html