Can I use ethernet instead of DMX for DIY led lighting?
You can do what you like; you could use the old Rank Strand analogue multiplex if you like (it's a joy to watch on an oscilloscope, for a start). The issue is whether you want your light units, whatever they are, to also be able to interface to DMX. If not, knock yourself out.

DMX may be more robust than ethernet in an electrically noisy environment, though it is a bit long in the tooth (it started coming in about 30 years ago, if the length of my teeth is a reliable guide). It's also designed to be transmitted over robust connectors rather than weedy RJ45s.

It probably depends on how pro you are, or want to be, or to look. Using a non-standard system is fine at first but if you invest a lot in it, then need to standardise, you're throwing a lot away.

NB I used to be a theatre lighting engineer but haven't been for a long while. Back in my day, real men used multicores with +10 or -10 control voltages.

edit
Bear in mind that interoperability is not just about interfacing to other rigs. You may want to purchase a unit yourself to add to your own rig, which would obviously need a standard interface.