Lactose intolerance is, as its name says, an intolerance, not an allergy, so the discussion about allergies do not apply
Lactose intolerance is genetic in its origins and originally all humans were lactose intolerant and in some countries like Japan, many still are. We developed the ability to tolerate lactose over the years when we started to preserve milk as a form of food, but many people still are lactose intolerant.
I have mild lactose intolerance. I cannot drink milk, or consume lightly processed milk products like yoghourt, cream and most soft cheeses, without feeling sick and my stomach being obviously inflamed for up to 24 hours. However I can eat hard cheeses like cheddar etc, with impunity. Another link www.sciencelearn.org.nz/resources/2016-genes-and-lactose-intolerance
My school days were made hideous by being young enough to be the recipient of free school milk and teachers would stand over me to make me drink it and then blame me for making myself deliberately sick. My mother knew I had a problem with milk, but mild lactose intolerance wasn't recognised then, I only discovered very recently that the reason I (and DS) cannot drink milk and lightly processed milk products is due to a mild intolerance.
On the other hand DD loves milk and buys the greentop unpasteurised if she can. The very sight of the bottle makes me feel queazy!