Update: Issues seems to have been resolved, you can now visit the
https://account.live.com/ChangeId.aspx url in your browser and sign into it with Microsoft account and you can rename Microsoft account from there (check the screenshot below).

Remember when you rename your  email address, your other account information, contacts and email don’t change. But it might take up to 48  hours  for all info to switch to new address. Read this Alias rename faq for more details. [Updated ended].

At present you can’t change your Microsoft account email address because this feature has some issues which Microsoft is working to fix. If you’re trying to rename Microsoft account email address of Hotmail to @Outlook.com and getting an error which can’t be changed, you’re not alone! this is happening to all and in fact Microsoft it self has intentionally disabled this feature since December 18 onwards because of some issues. You’ve to wait till Monday that falls of 7th of this month to know about the rename feature’s fate.

Microsoft Temporarily disables Account Renaming feature

A thread in Microsoft community forum says: when you make an effort to change Microsoft email address you’ll get the message “The email address associated with your Microsoft account can’t be changed” .

This issue exists from December 18th where Tommy of Partner, partner technical head for Microsoft says “the rename feature which allows you to change the email address you use to sign into your Microsoft account is currently not available. Our engineering team has identified several issues with rename and has temporarily disabled the feature while they investigate.”

He confirms they’re working on it and you can know the update on it on January 7h of this month.