The LIghtning Network is not generating any coins. It's locking funds between two (or more?) parties (on chain) so they can transact between themselves and settle on chain at a later point.

Proof of work is doing work to win the lottery to get the privilege of furthering the blockchain with the added benefit of allowing the winner to reward themselves with coin.

Proof of stake is staking already generated coin to win the lottery to further the blockchain (and give themselves a reward).