What are the options for milk?

You could certainly braise cabbage in regular or oat milk, but it isn't going to reduce to give you the same final "pudding" like consistency. You get that because of the nut solids which are left in the "milk". I don't know if oat milk would react the same way, but suppose it could since oat milk would contain some of the oat fiber and such. Your flavor profile will also be different. Regular milk is going to end up tasting a bit carmelized, and oat milk is well, oaty. And depending on the fat content of your milk, you could end up with some curdling from the heat. I'd stick with the cashew milk, or perhaps another nut milk, myself.