For hole saws that cut metal, this is all I could find.These are for knockouts in electrical work but would pop right thru a sink I think. They will cut #10 gage steel. That’s about 1/8 thick.
A good quality hole saw made for cutting metal will also work or the step drill Neal posted above.
DO NOT use that wood bit you posted the photo of.
.I've never needed a 1.5" hole for a regular kitchen faucet let alone an R.O. Usually 1 1/4" is more then sufficient. Do me and yourself a big solid and size the hole up with whatever you decide to make the hole with. You can always make the hole bigger but smaller is more difficult.
Puddlesx5 is pointing out that something don't look right in that picture. The part that goes through the hole would be expected to have an external thread for a plastic nut. Photo does not show side very well, but it don't look threaded.You only have to measure the bit that would go thru the hole.