Could I have a milk allergy or could it be something else?

Potential milk allergy?

I'm 19m, this has been happening since I was a toddler. If I drink pure liquid milk, I get an itchy rash on my arms, especially in my inner elbows, sometimes with a few clear fluid-filled bumps. It usually appears about an hour or two, sometimes a full day, after the milk. It spreads and becomes more irritated if I drink more than a single glass. However, I have zero reaction to cheese, most ice creams, creams (sour cream, whipped cream, crema, etc), butter, and yogurt. And only some mild stomach upset for milkshakes, milk based drinks in general, certain types of ice creams (mostly soft serve for whatever reason), and buttermilk (this one actually straight up makes me physically gag and sweat even in small quantities despite me not minding the flavor)

If necessary: I'm 4"11, 100lbs, latin American, and have all of the symptoms of thyroid disorder + a family history of it, however it does not show up on tests, and I have generally been a sickly creature all my life