Calling Lightning a Proof-of-Stake system is orthogonal to the concept usually labeled as PoS. Him having done so will just generate loads of confusion.
What we usually describe as PoS is a mechanism to achieve distributed consensus among a network of participants. Staking there is providing updates to the whole network's state, where you earn fees for contributing to the validity and the security of the network.
On the other hand, Lightning Network is a system of bilateral smartcontracts, where you may earn a fee by providing liquidity to other users. Nodes along a payment path don't provide any proof whatsoever, they merely forward a payment. There is no global consensus in Lightning, contracts are created, verified, and executed between two or a small number of participants. Any need to create consensus in the case of disagreement is provided by the proof-of-work of the Bitcoin network.