EDIT 17OCT25: With the latest Battlenet update, this is now working as intended.
This has only been happening for the past two days.
Each time I open Battle.net, it takes me through the entire process of entering my username/email, password, tick the password remember box, enter/accept the Authenticator code and log in.
Closing Battle.net starts this process all over again - my login information is not being saved even though I have selected the ‘Keep Me Logged In’ checkbox.