This may be a very niche issue, but I spent a lot of time on it and wanted to share my solution. I have a database on an Azure SQL Managed Instance with spatial data (it's not an Enterprise GeoDatabase so there's no SDE schema). I wanted to set up a connection to it from ArcGIS Pro 3.5 using Microsoft Entra Integrated authentication. I had no issues connecting this way from SSMS but when I tried in Pro I would get “Database Connection: Bad login user[ ]”. During troubleshooting we found that things worked when my user was made a sysadmin on the database server but no other combination of permissions would work. Finally, I upgraded to ArcGIS Pro 3.6 and it worked so the issue appears ultimately to be a bug/lack of full support for Entra authentication in ArcGIS 3.5.
@ChristopherEby2 - interesting. Question for you - what is the character length of the username that fails at 3.5? At the 3.5 release, the limit was 30 characters and at 3.6, that restriction is no longer present. Further, there was an ENH-00121539 - "Allow ArcGIS Pro to handle user names, schema names and database names longer than 31 characters for DBMSs that support this functionality." that was implemented at 3.6 so perhaps that's why it works at 3.6 for you...
The username was 30 characters but it sounds like that was short enough. Even when I was getting the "Bad login user" message, I could see in the ODBC logs that initial authentication was working and the failure was happening when ArcGIS Pro queried the database for schema information.