When I milk on the trail it is as low tech as can be. Milk filters are very lightweight and can be burned after they are used.
I milk into a small stainless pot with a milk filter rubber banded over it, the milk goes right through and then I put that milk, warm, right from the goat into oatmeal, my tea, cup or cold cereal. Sanitary is not possible on the trail, and considering the fact that I literally am touching my goats and then feeding myself gorp while hiking.. I mean, um, bacteria ARE getting into my system. Haven't you seen how gross your hands get on the trail? My goats kiss me on the lips for goodness sakes, germs are the least of my worries!
Nothing has time to go rancid, it is milked and consumed right then and there. There is no refrigeration so storing it is pointless and making cheese on the trail is a bit cumbersom, but yogurt is a neat thing to make on the trail if you don't mind hiking all day with a quart of it brewing in your pack! Our dogs like the extra if there is any!
I've also milked one handed into a stainless canister (narrow top) and then stained it directly into the oatmeal pot or whatever. This takes a couple of trips as the canister is only about a quart, but it keeps the milk clean. Bringing this along, one stainless canning funnel and a sandwhich baggie of milk filters is not a lot of gear. As for washing the udder, well, I bring some biodegradable wet wipes or just plain paper towels, get them damp with drinking water and then just rub her udder and belly with that. These are also burned or dried and then packed out, like the milk filters. Truly, there is not a lot of need for anything else. If you must bring teat dip, it can be kept in a small container with a lid but it would not weigh much either.
I guess if you were clever about campsites, or lucky, you could find a "stanchion" log, slab or rock, but I am fine just squatting down on a surface that is not dusty. My does stand fine for this and if they are figety, someone holds them. They've all been taught to milk without grain but afterward I give them a handfull of sunflower seeds or calf manna and they have considered it a good trade. No need for anything else except they need more browse time for the extra calories. I haven't hiked with a rail thin milker though, my two have been a good weight.
The milk stuff is washed like everything else on the trail, with a little bit of soap and then rinsed in hot water and allowed to air dry. Nothing special. We are eating dirt the entire trip anyway.
We call it, "priming" our immune systems. And, if you want to rag me for it that is fine but I find it hard to believe anyone keeps their hands sanitary out there, especially with goats along who like scritchies at rest stops. Seriously, are you telling me that you've never dropped a cookie in the dirt in camp and then picked it up, dusted it off and eaten it anyway? Really?
I have.

Charlie Goggin
Lightfoot Packgoats