Lightning only has similar sender privacy guarantees IF AND ONLY IF you're running your own LND node with a channel opened to an onion peer with a UTXO that's untraceable to you.

Using a custodial wallet like Wallet of Satoshi? No privacy. Phoenix or Zeus with Olympus? Little privacy from the LSP, the metadata makes timing attacks trivial.

Only exception being ecash but it's custodial and anonymous nature makes it extremely susceptible to counterparty risk at the mint.

"Lightning has awesome privacy" is true only in the perfect case. For most they have worse privacy than even the noob Monero user that uses a remote node with no VPN with coins purchased from a CEX.

Let that sink in. The worst case scenario privacy wise for Bitcoin, is superior to all but the most schizo levels of Lightning Bitcoin usage when using Monero.

That is why Monero is used on darknet markets and lightning isn't and on-chain Bitcoin is all but ceased. Monero is private even for the noob, Bitcoin is public for all but sysadmin Linux expert types.