Sound like you got US-Keyboard loaded even when a german keyboard is selected, because on the ö-Key is on the US-Keyboard the ;If i type this keys I get different letters e.g. ; if I typoe ö.
The language in the settings is correct!
Greetigs Reggie from Germany
Do you get a Z on the Y-Key?
The DE-Sign is only in the upper right at the Login-Screen.
Did you check the Chrome-Browser Settings after the Login?
Einstellungen->System->Sprachen und Eingabe
Did you use another system with chrome where the Keyboard could be in the US-Setting and this would be synchronized via the web to your chromebook
Youd could try to switch between the installed Keyboard-Layouts:
Alt+Umschalttaste -> vorwaerts
(so zu sehen in der obigen Sprachenkonfiguration)
Nice Greetings from Germany near Frankfurt/Main