I have created a Mojang account with a password. The password includes the Umlauts "ä" and "ü".
Logging into Mojang works fine.
However, logging into minecraft.net or the game Minecraft on PC results in an error, that the password is not correct.
Changing the password to not include Umlauts fixes the issue.

You should either not allow Umlauts in Mojang passwords in the first place, or support them on Minecraft.