When your Microsoft account works in the browser, does it open an existing signed-in session, or can you enter your password and sign in again successfully? Also, which Windows 11 Settings page produces this prompt—Accounts > Your info, Accounts > Email & accounts, or another page? The unfamiliar address is being presented as a verification destination, which can differ from your account’s sign-in address; the prompt alone does not establish whether your sign-in address was changed.
One useful additional check is to open Manage how you sign in to Microsoft from the browser session that currently works and check whether the Gmail address displayed in Windows appears under Account aliases. Microsoft allows several addresses to belong to the same account, so comparing this list can help establish whether Windows and your browser are accessing the same account. Please tell me whether that Gmail address is listed, missing, or whether opening the page itself requests verification through the unfamiliar address. You only need to report the result; please keep full email addresses and verification codes private.