I had the same issue for a long time. Today, I have switched to "Linux Mint 21.3 Cinnamon" and installed Edge by downloading ".deb" file from official page.
I could add my first Microsoft account as normal. I have clicked to "Set up new personal profile" and added a new profile.
SS: Set up new personal profile
When I try to sign in to my second Microsoft account it opens a floating modal to enter email.
SS: floating modal to enter email
After I enter my email it switches to the other profile (my first Microsoft account) and asks me to enter password there.
SS: auto-switch to first profile
Even if I enter my password, it doesn't forward the authentication to other profile so it gets stuck.
SS: stuck on sign in state
SOLUTION:
Go to profile settings of new profile.
SS: profile settings button
Click "Profile preferences".
SS: Where is "Profile preferences"
Here I turned off everything (profile switching options, auto sign in, allow single sign-on, etc.).
SS: Switches to turn off
Additionally, I did the same (turning off profile switching and other settings) on my first profile and set the "Default profile for external links" (Profile preferences) to the new profile.
Maybe not all these steps are necessary, but you may just do them temporarily and reset settings after you finish.
Then I could just sign in my second account without any profile switch. There might still be a problem if you are using Microsoft authenticator app to sign in without a password, but it works when I use password. In my case, I have used password and one time code (OTC) from the authenticator as 2FA.
Hope it helps :)
