If you selected standby, the database should be in the standby state. Recovering is when you select the NoRecovery status.

Standby's only if you need to read the database for any reason, if you don't need to run reports on it, leave it in standby. If the logs are restoring (and that you can see in the error log or MSDB tables) then the secondary is up to date as of the last log it restored.