I have the same problem. I started getting dialog box Windows Security asking for password and not remembering it after reinstalling Outlook 2013 Hotmail account on a new Windows 11 machine.
Same outlook account was working fine on Windows 10 machine before. Now my Hotmail account asks for a password every restart. It even keeps asking after that. Sometimes it works and gets mail, sometimes it doesn't with no error message. It is very strange.
I have been using hotmail (currently outlook.com) from browser and was happily using it from Outlook 2013. After reinstalling the same acoount on Windows 11 I started having this problem. No matter what I tried, I could resolve this issue. Why does Outlook keep asking for password and not store it even if I check remember checkbox.
I realised if you to to e-mail accounts setting and try to repair my hotmail account it does not work. I you try to set it up again manually, you get a dialog box: Microsoft Exchange with tabs: General Advanced and Secutity. In the Security part there is a checkbox: User Identification - Always prompt for logon credentials. This checkbox is grayed out. It seems unchecked by default. You can get to this by not running outlook and using Control Panel > mail icon, account settings. I couldn't modify it.

Any suggestions?