"Can you only be allergic to cow's milk?" Short answer, yes.
It depends on the individual. For instance goat milk causes me less trouble that cow's but I still get sick from goat milk. My dairy intolerant niece however, has no trouble with goat's milk.
Milk has two proteins, one is casein, the other is whey. I tolerate casein ever so slightly better than whey. I discovered that I can eat a tiny bit of cheese if it is very aged like Parmesan or extra sharp cheddar because the whey is almost entirely gone - but even at that it has to be a small amount or I'm back in trouble with a nasty stomach ache (or worse) again.
I've also found that my ability to tolerate certain foods can depend on my overall health, if I'm feeling more run down I seem to be more sensitive to certain foods than I am when I'm feeling more up to par.
Hope that helps
If you are looking for the 'something more,' you might want to check out Celiac Disease, as well. It involves reacting to wheat, rye, or barley, but many people are asymptomatic. However, it damages part of the intestine that is involved in milk digestion, so it can CAUSE issues with milk that go away on a gluten free diet.
Also, it can trigger at any age. 1 in 133 people have it, in randomized studies. People with this disease are more prone to food allergies. And, sadly, in a recent study, over 60% of Gastroenterologists surveyed, world wide, did not have accurate information on important aspects of diagnosing this disease. Symptom recognition was often one of the big lapses, because so much has been discovered recently that many doctors are not yet aware.
And just as an aside, on the whole casein and whey front? Those are actually two groups of milk proteins that include a number of proteins each. So lots of different proteins that can be reacted to.
On the milk front. If one is lactose intolerant, milk from mammals, period, is an issue (except camels, because they have a rather different milk composition that seems to alter how we react to it.). If it is an allergy to milk, then other milks may be okay. However, many other milks have a higher incidence of cross-reactivity with cow's milk - in other words, there is a higher likelihood that your daughter will develop an allergic reaction to other milks with a higher incidence of cross-reactivity. Goats milk is one of the ones she may be likely to develop a reaction to. I believe sheep's milk is, too.
One question for you: does your daughter react only to the consumption of milk, or to touching dairy as well? If touch is an issue, then it is absolutely an allergy. And if touch is an issue, you'll need to watch out for latex - they add dairy to that, usually.
You'll also want to be careful of apples, citrus, and pears. Most of the time, these fruits have wax coatings added (shellac wax) and this typically has either soy or dairy added to keep the texture right. If your daughter is really sensitive, obviously that'll be an issue. I'm not sure what other fruits and veggies may use shellac wax coatings, but there may be others. The other waxes don't use dairy additives, to my knowledge.
From reading elsewhere, there are three distinct conditions: lactose intolerance, milk allergy, and milk protein intolerance. On the last, there's some disagreement about whether it's a sort of allergy, or a digestive problem about breaking down milk proteins, similar to lactose intolerance being a digestive problem about breaking down milk sugars. Lactose intolerance sometimes follows milk protein intolerance. Milk protein intolerance is difficult to positively diagnose. But it tends to be a slower reaction than milk allergy, which is often quick, or lactose intolerance, which follows within a hour or so. Milk protein intolerance might come on a day or two or three later - and thus be more difficult to connect with the milk exposure.