log ship from older version to newer one
As the error message mentions, you can't use WITH STANDBY when restoring to a newer version of SQL Server.  The database upgrade process has to run in that case, and it can't do that in standby mode because the database is essentially read-only.  This is documented in RESTORE Statements - Arguments (Transact-SQL):


  STANDBY is also not allowed when a database upgrade is necessary.


You can use NORECOVERY rather than STANDBY for the restores instead.  You won't be able to use the new server as a readable copy of the database, which might be a dealbreaker.  But it supports the scenario of migrating to the newer server (if that's what your end goal is).

I would double check your configuration. Specifically, check that you selected "No recovery mode" for the "Database state when restoring backups" option on the Restore tab for the target (secondary) database.  It's step 17 in Configure Log Shipping (SQL Server) - Using SQL Server Management Studio


  
  On the Restore tab, under Database state when restoring backups, choose the No recovery mode or Standby mode option.
  


You can review the settings by right-clicking the database, choosing properties, and then looking at the "transaction log shipping" tab.

Clarifying a few questions and statements from the comments:


  It seems that at the time of database open on 2016, it actually applies/restores transaction log files. Please confirm!"


No, that's not how it works. The transaction logs are applied during the RESTORE LOG operation. So the deleted source transaction logs aren't a problem. If you're getting that error about missing transaction logs, then it sounds like some of the restores failed, or they were restored out of order, etc.


  However, when we check the timestamp of data and log files of the configured databases on 2016, it shows the timestamp when this was configured 


You can't use Windows/NTFS timestamps as a valid indication of the SQL Server data files being used, and thus it can't be used as evidence that the transaction logs are not being applied when you restore them.