The short answer to your question is... sort of. In general, the nutritional requirements of different species do vary, even for infants and so the composition of the milk is important.
Consider, for example, that cats are obligate carnivores and require certain nutritional elements in their diet for health and growth, notably taurine. In addition, many cats are lactose intolerant and, while kittens do have lactase in their system, the amount of lactose in cows milk (to reference the assertion from the comments) is way more than the lactase in the kitten can handle. Add into that is that the proportions of other compounds, such as casein to whey, is off for kitten nutrition.1 Milk replacer for kittens does start with a bovine dairy source, but it's been modified and enhanced so as to be nutritionally appropriate (as it is for human infants).
That's not say it's impossible to cross-species nurse, by any means, but that in order to do so, you want to confirm that the make-up of the source milk contains the appropriate dietary support for the infant animal or that you're able to further supplement with other sources of nutrients. Bear in mind that cross-species nursing in captivity is often done under the supervision of veterinary specialists.
In the wild, this also does happen, surprisingly often, and some species tolerate it more than others on the giving front. In that event, the infant may get just enough solid nutrients to survive to the point that they're able to eat more adult food and continue to survive. In such cases, I would expect there to be some additional risk of reduced resistance to species-specific natural illnesses and possibly weakened bone or musculature, but that's just a risk, not assured. The success of this, long term, is likely going to work best when we're talking cross-species with comparable diets, carnivore to carnivore, herbivore to herbivore, etc. Not to mention that competition for the milk probably works best when the outside species is relatively able to compete for it.
Finally, on the tl;dr front -- Cross-species nursing is going to be a case of better that than nothing at all. If the choice is the milk of another species or starving to death, then the choice is rather easy to make...
1 Cats and Milk: Lactose Intolerance and other Dietary Concerns