Yep! More private than monero for a lot of reasons:

- lightning invoices reveal less information than monero addresses

- in lightning transactions, the sender is encrypted (unlike monero, which uses 15 unencrypted decoys and 1 unencrypted real spender, and just hopes analysts can't eliminate the decoys)

- in lightning, the full amount is encrypted (unlike monero, which leaves the fee in plaintext)

- in lightning, the recipient is encrypted (unlike monero, which typically leaves the recipient's address and a change address in plaintext, though it does randomize their order, which is good)

- none of that data is published on a blockchain (unlike monero, where it is)

Monero does some pretty good things for your privacy and I can only identify the "real" sender of a monero transaction in about 1/5 cases, and only with about 80% confidence. Chain analytics companies can do a lot better because they have off-chain data that helps them eliminate more decoys. But by comparison with lightning, monero is pretty transparent.