It's not a good argument which doesn't meant hat there aren't better arguments in favor of it. The problem with "we're the only animals that do X" arguments, is that you can substitute for X a lot of other things besides drinking milk and then it becomes problematic in an obvious way. E.g. we're the only animals that walk upright, we're the only animals that cook food, etc. etc.
Now, there is something to be said about dairy products not being the ideal source of calcium and protein due to containing saturated fats. But they are rich in protein and minerals like calcium. If we leave out dairy products from the diet, then it seems to be quite difficult to get to the RDA of calcium. But here we must take into account another difference between humans and animals; humans are sedentary animals whose typical fitness level falls way short of the maximum fitness level they could achieve.
Now, I do exercise a lot, I eat close to 4000 Kcal a day, and I do get a lot of calcium from sources that would be irrelevant for most other people. These sources are:
Water contains 60 mg of calcium per liter, I drink 3.5 liters a day, so I get 210 mg from tap water alone.
Whole grain bread contains 30 mg of calcium per 100 grams, I eat 500 grams a day, so I get 150 mg from dry bread alone.
Vegetables, some contain a lot of calcium, some contain oxalates that block calcium absorption, so one has to be careful with making estimates here. An example could be broccoli, this contains 47 mg per 100 grams. I typically eat 500 grams, so I get 235 mg from broccoli.
Potatoes contain 12 mg per 100 grams, but I eat 1 kg of potatoes, so I'll get 120 mg from the dry potatoes alone.
So, an a typical day I can easily get to (210 + 150 + 235 + 120) mg = 715 mg of calcium (the RDA is about 1 gram) from sources most people wouldn't even count because "they hardly contain any calcium". Now, I do also eat cheese, yogurt and other dairy products. Just two slices of cheese contains 600 mg of calcium.
Most people eat only half the calories I eat, this would bring down the amount of calcium they could typically get to about 360 mg. However, most people also tend to get a smaller fraction of their calories from whole grains, and more from fat. Instead of eating a sandwich with meat, they eat meat with a sandwich like this:
Then you can forget about getting even that 360 mg of calcium from non-dairy sources. A few hundred mg of calcium is all most people can count on from getting from non-dairy sources, therefore they can't do without dairy products.