I was running into the same problem, and it was because I had logged in using my Email Address rather than my User Principal Name (UPN).
Some organizations (mine included) give each user an Email Address that is different from their UPN, and while most SSO login dialogs will allow you to login with either one, apparently some Azure-based SQL endpoints only recognize the UPN as a valid login.
To authenticate with my UPN instead of my Email Address, I clicked the little "person" icon near the bottom of the menu bar on the left side of the Azure Data Studio window, then clicked the "plus" button on the "Azure" bar in the Linked Accounts pane. When the SSO dialog popped up, instead of selecting my usual Email Address, I clicked "Use another account" and entered my UPN and password. Once I did that, everything worked as expected.
Hope this helps!