Can a Miner or Mining pool have exclusive access to a transaction and its fees?
A non-standard client could choose to send his transactions only to a specific miner. One example might be for a client to negotiate a "monthly package" (off the blockchain), and then send all his tx directly to the pool, without any specific tx fees.

There is nothing in the protocol that prohibits or enables it - it is a possible extension that is outside of the protocol. To my knowledge, no such schemes have been reported in the real world.