After a SQL Server availability group failover has taken place, an existing SDE connection to that availability group gets disconnected in ArcCatalog. Get "Microsoft.SQL Server Native Client 11.0" Communication Link Failure error message
I think that is expected behavior since the Enterprise Geodatabase (EGDB) connection is "severed". Each connection has an assigned process ID that is specific to that session.
The benefit of the fail-over option is that when the primary goes into a failed state you can reconnect (restart the session) and keep working on the backup EGDB. Now ArcGIS for Server will automatically fail-over when the connection is dead. This is because the service will continue to check for a connection once lost, if no connection can be made to the primary it will fail-over.
Connections to highly available SQL Server databases—Help | ArcGIS for Desktop
If the principal server becomes unavailable, ArcGIS for Server retries the connection automatically. At that time, if the mirror server is available, the service's connection will switch to using the data on the mirror server.
Hope this helps.