Managing personal Outlook.com account settings, security, and privacy
Microsoft has removed the ability to add a phone number to a personal Outlook/Hotmail/Microsoft Account.
It's gone. Not there for anybody. You can remove an existing phone number but you can not add a new one or change an existing one. "By design", not a bug. There is no way around it.
Not at Ways to prove who you are (Verification)
https://account.live.com/proofs/manage
Not at How you sign in to your account (Username/Alias)
https://account.live.com/names/manage