Configuring preferences and options in Outlook.com
This solved my problem. Thanks. The Save button appears after I have added a phone number to my account!
This is it!
I actually had to add a phone number to the 2 step verification part of my* account. I suggest whoever has this problem to try and add new methods of authentication. Outlook doesn't tell you why you can't add forwarding there but, at leat in my case, it had to do with adding a phone number to the Account Security section of my outlook (web).
I'm not 100% sure but I have a strong feeling it would have to do with the security info of your own outlook account in almost all cases since, according to what I have read in this forum it is not a cache problem.