I made intensive tests to narrow down the problem. Here my results.
I went to php, php configs and used one config. I changed the switch of "Übergeben von HTTP AUTH BASIC/DIGEST-Headern von Apache an PHP" to on and saved. I called this configuration, and the switch was off. I checked the same with "Für aktuelle Kunden automatisch hinzufügen". Switch stays off. I changed other information in the php.ini Einstellungen: they correctly saved.
I checked this with other php configurations. Same behavior. I used other Browser (instead of Firefox, Chrome) I checked it on other computer. Same behavior. From my point of you, it has to do with the Switch on/off elememt. So I checked next the behavior of the switch in "Domain bearbeiten". There switches work like expected.
So for me it means, the switches on the PHP-Config page are the reason of the problem. I have no idea why. I will investigate the problem further with clientsite debugging. Maybe you can give me on basis of this information another hint. I would great appreciate your help.