Is it a milk allergy instead?

Milk allergy symptoms are similar to symptoms of a lactose intolerance, but these conditions aren’t the same.

A milk allergy is a type of food allergy that occurs when the immune system overreacts to dairy. If your baby has a milk allergy, they may have an upset stomach and diarrhea. But they’ll also have symptoms that don’t occur with an intolerance:

wheezing
coughing
swelling
itching
watery eyes
vomiting
If you suspect a milk allergy — even a mild allergy — see your doctor. A milk allergy can advance and cause severe symptoms like a drop in blood pressure, trouble breathing, and anaphylaxis. According to Food Allergy Research and Education, milk allergies affect about 2.5 percent of children under 3 years old.

https://www.healthline.com/health/baby/lactose-intolerance-symptoms-in-babies#vs-milk-allergy