My son was the same way. As my husband said "Why drink from a cup when you can get it straight from the tap. It's always better from the tap."
Is it that he doesn't like cow's milk or doesn't like the sippy cup ? My daughter never took a sippy cup. She went right from breast to a free flowing cup. We took out the no-spill valve inside to make it free flowing or just used a regular cup and helped her hold it.
I might suggest trying something different. If he is breastfeading cows milk definitely tastes different and if it is cold that will make a difference. We gave our son rice dream (better for digestion than cows milk) and the vanilla has a bit of sweetness to it. We just warmed it up a bit and he loved it.
I don't blame him ;). Your breastmilk is all he needs, so I wouldn't worry about it. He WILL take a cup, I promise, but I'm sure it will be on his terms. My son took to a Safe Sippy first. The top is very "boob-ish". Good luck!
If it is just the taste he doesn't like and not the actual sippy cup part, you can try what I did with my son and slowly add more whole milk and less breast milk to the cup. For example we would do 1/4 whole milk and 3/4 breast milk and every few days would do a 1/4 more of the whole milk and 1/4 less of the breast milk. He didn't like whole milk at first but when we did this he never even noticed the change!
My son wasn't a fan of whole milk at first either, and I was still breastfeeding him at the time that I started giving him the whole milk. I did a few things -- first I put the milk in a clear plastic glass so he could see what it was (prior to that I was using a sippy cup and I think he was so used to getting water out of it that he wasn't sure about the milk). Then I would pour myself a glass and let him see me drinking it. I also added a tiny bit of Nesquick to his milk. Since he was at the stage where he wanted what we had, it didn't take long before he wanted to try the milk himself. Within a few weeks he was drinking it on his own, in his sippy cup, without the Nesquick.
I wouldn't worry about him drinking the milk if he's eating cheese and yogurt. The breastmilk is better for him than the cow's milk anyway. (Easier to digest, has more species-specific substances.) When he starts to nurse less, his thirst will motivate him to get it from a cup. I've noticed my daughter drink more as she nurses less.
Don't force dairy - humans are not designed to drink cow's milk - your child may instinctively be avoiding a lactose intolerance - look for other natural foods that can provide the vitamins and minerals your baby needs.
My son was breastfed (still is) and he hated whole milk at first. He is just now starting to like it and he is 19 months old. I didn't give up on him. Every time I gave my oldest son some milk I got some for my little one. Even if he spit it out every time or drank one sip then spit it out I kept trying. He only drinks a couple ounces a day but at least he is starting to like it now. I am hoping he will eventually wean himself and drink mostly whole milk but as long as he is happy I'm not too worried about it. Your son is getting most the nutrients he needs from you breast milk still. My son drinks better from a regular cup than any sippy cup. He will take a sippy cup but he usually won't drink as much from it as he does a regular cup. It took a couple months but he got to where he is actually better at drinking from a regular cup than my oldest son sometimes.
It sounds like he's getting plenty of milk. Maybe you can start in a few months with a sippy cup of water or water with a little bit of juice in it. I nursed till my son was about 20 months (the last few months was only once or twice a day) and never gave him any other milk. The doctor at his 2-year appointment said that dairy 3-5 times a week was enough. Anyway, that's much less than I expected to hear. I don't think you need to worry about it.
Contrary to what doctors and the dairy industry will have you believe, kids don't NEED milk to thrive. neither of my children drink it, they both hate it. The milk out now is pretty much junk food, all the good stuff is taken out with pasteurization. They do drink rice milk at times, but since many foods are fortified with calcium and there is Vitamin D from the son..we aren't too worried about it. He's NOT nursing too much either BTW...mommy milk is made for humans and has a far different taste and fat and nutrient content then cow milk. Try other types of "milk" and see if your son is interested in those, or you can keep trying the milk off and on, but just know he will thrive on your own milk, even if he does not drink the other kind.