Should replica DB be in 'recovery' - PostgreSQL

If you set up Postgres replication by sending write-ahead log as described in streaming replication manual, then yes, all replicas are in the permanent recovery state which is normal. As write transactions are performed against the master node, the replicas receive and apply logs from the master, as any Postgres server in recovery would do.

The changing number you see is the currently received transaction ID.

Please read the complete linked page to clarify details. As for me, this is clear that the standby (slave) server actually performs recovery continuously unless promoted.