My question is: is it just part of the paint job, or does this yellow square serve a purpose? I haven't seen this square on any other airlines.
It's a yellow square that says Willkommen / Welcome. In one case I have seen that additional text was written into the square with a black (removable?) marker.
This website explains it like this:
A yellow welcome panel painted beside each of the passenger boarding doors is now visible on each aircraft. The mark is a flat yellow rectangle, with the word 'welcome' in both German and English, accompanied by the iconic logo in the corner. What I love so much about this - is that that bright yellow warmth is presented to every passenger, right as they board every aircraft. Upon boarding, each passenger is welcomed, setting the stage for the onboard experience and hospitality that will follow. The crew can write personalised messages to give a more human welcome to this otherwise mechanical setting.