I've never heard of Hashimoto's causing lactose intolerance BUT there seems to be a strong correlation between Celiac and casein intolerance. Lactose is the simple sugar of milk and casein is the protein of milk. It does make sense if you think about it.... If Celiac disease is an autoimmune reaction to gluten, (the protein in wheat) then it makes some sense that the protein in milk might also cause issues particularly because both proteins are literally sticky and bind in the small intestines.
I bring up celiac because there is evidence mounting that suggests that there might be a genetic link between those who are susceptible to getting celiac or at least gluten intolerance and those with hashimoto's and other autoimmune disease. The gene they are looking at is HLA-DQ2.
Its not medically documented. But I will say that we see Hashi people that do better eliminating lactose over on the thyroid forum. I switched to almond milk without soy, tastes great, good for you too. Hashis and soy is a big no no, despite what 'soy' people say - they seem to know minimal about Hashimoto and hormones.
One thing about milk. All mammals (us included) by nature, are supposed to wean off milk during infancy. Humans have just decided not to.