If an owner releases a player in the middle of a contract, for example, in year two of a 5-year contract, 12 million per year. I understand that the owner is responsible for 99% of that contract for that current year. What about the following season? Who is responsible for the 12 million in the rest of the contract aside from the existing season he was released in?

If he is released then the owner releasing him is responsible for the remainder of the contract. If he was waived and claimed off waivers then the team that claimed him would be responsible for the remainder of the contract.

Yeah just to make it absolutely clear: when you outright release a player the remainder of his entire contract STAYS on your salary cap - now and any future seasons.
Any other team which picks that player off waivers gets him for the veteran minimum, 378k per season.
As Chilly said, if you want to get rid of him you can try waiving him and if any other team claims him they take on his contract. If they don't you are still stuck with him.

So it might be wise to waive said player towards end of season when contracts are prorated and less costly to pick up for that season. A claiming owner who doesn't examine contract details in that type situation is essentially claiming full cost for future seasons. This is a strategy one could use to rid the team of a crappy contract towards season's end (as long as the waiving team is out of contention and player has minimal impact on win %)