Unfortunately, it has nothing to do with the pasturization process and, in fact, there are super-processed milks available for people who are lactose intolerant - nearly everyone over five years old. We are not able to process the milk of any animal (including humans) after about the age of five because all of our nutrition should be coming from other foods by that time and we don't need to nurse anymore. Mommy's milk is great for babies, but even by the time they are a year old, it's just not as useful. Once we reach an age where we would naturally stop being dependent on milk for nutrition, we stop producing lactase, the enzyme that digests milk.
Well... most of the world's population stops producing lactase. If you are of European, especially Northern European heritage, you are much more likely to be able to digest milk even as an adult.
You are right about milk from other animals though. It would be dangerour or even deadly to try and raise a nwborn human on cow's milk, because the nutritional content and types of sugars are completely different. Same thing goes for trying to
feed an elephant milk from a sheep.
Yes, this is rather a long reply, but I hope it answered some of your questions. You can google the subject too if you don't trust my uncited statements.