No but the breed can affect how much you get, how it is fed can affect quality and flavor somewhat as well. This why there are breeds that are considered "dairy" cows and breeds that are considered "beef" cattle, the beef cattle breeds do not give as much milk or as consistently. Dairy cows give lots of milk and for longer than beef cattle but their meat is not as plentiful or good as beef cattle.
Not the color but the breed. Those black and white cows that most milk comes from are called Holsteins. There are other dairy cows that are brown called Jerseys and others called Guernseys. Jersey milk has more cream and a richer flavor. Guernsey milk has more cream than Holstein but a little less than Jersey. Guernsey has a little bit different kind of lactose and it has Beta-Carotene and more protein. My favorite kind of milk is Raw Whole Guernsey. I wish I had a big glass of it right now. Mmmmm good!