Inherited Permissions: SQL Server 2005 vs 2016
I would not expect the different SQL Server versions to make a difference here. Although the setup might appear identical, there are likely subtle differences between the databases on the 2 servers or the server config. 

The first thing I suggest is to verify requirements cross-database ownership chaining are met:


Ensure the 'cross db ownership chaining' configuration option is the same on both servers
Make sure the DB_CHAINING database option is identical
Check that the database owners are the same to maintain an unbroken ownership chain for dbo-owned objects


If the culprit is the 'cross db ownership chaining' server option, it would be better leave that off and instead turn on the more granular DB_CHAINING database option for the SQL 2016 databases in question.

Post the actual error message if you still have problems after the above.