Working with two Microsoft SQL 2012 servers in Always On High Availability configuration running on Windows Server 2012 R2. Setup works well and failover is almost flawless.
However, the software that connects to database (or high availability cluster in this case), is using sql user account that must be mapped to database. I have created said user on primary node and mapped it to database. On second node I have created same user but am not able to map it to database. Because of that, when failover happens the database is technically available but software can't connect because user is not mapped to database on second node.
I assume this is the issue because I am able to connect to high availability after failover using windows authentication.
I have ignored, or rather forgot, the fact that sql user accounts on servers had different sid which in fact was the source of the issue.