The problem is the blockchain fungibility: if you have a non-fungible blockchain, the lightning nodes with the biggest liquidity will be easily identifiable, and thus susceptible for regulation.
I don't think a regulated exchange or payment processor will just connect to whatever other Lightning Node without doing extensive KYC/AML checks. So this means that when you want to deposit BTC in a liquid Lightning Channel, you better own "white" bitcoins .
Your "black" bitcoins will be even more difficult to move due to (1) presumably (relatively) less transactions on the blockchain itself and (2) mixing on the blockchain will not remove the taint of your coins and (3) due to the blocksize limit, it is expected that with increased transaction , (mixing) transactions will become more expensive.
So I would argue that Lightning will DECREASE fungibility in stead of increase it. To conclude, let me quote my good friend u/andreasma (https://youtu.be/4w-bjUhpf_Q?t=3m42s)
What is the important thing we need to achieve in the base layer is not scaling; the important thing we need to achieve in the base layer that CAN NOT be achieved in the layers above is fungiblity and privacy with strong guarantees and simple primitives. And if we have privacy primitives and fungibility primitives in the base layer than we can do scaling in the second layer and we can do it securely. Otherwise, we have a privacy problem. And that privacy problem will get magnified as we go up the layers. If you can do analysis on the base layer, that gives a great degree of insight into what's happening above."
In contrast, when you want to run a Lightning Node on Monero someday, it'll be hard to identify the owner. The big exchanges could self-identify their nodes, but there will probably still be a good part of the network that is unidentifiable. So LN can potentially decrease the fungibility of XMR a bit, but we always have the option to go back to the blockchain (thanks to the dynamic block size), do a few transactions and get nice fungible coins back and use them again on LN if we want.