I'm 35 and have had the same issue all my period life. It's not the milk. It's the sudden hormone change. It can be really rough on some peeps. If you are on a bc you may consider changing to a different type. If you're not on one at all you may consider trying one to balance things out. I was using the mirena and found out I'm overly sensitive to progesterone and had it removed. Doing much better now, however, I had to go old school and get a diaphragm because nearly every single bc has progesterone on it. Good luck.
It would be odd for it to affect your period unless you normally only drink milk around the time you have your period. Of course, if you notice problems when you drink milk in general, then you may consider leaving dairy out of your diet and see if that helps. There are plenty of non-dairy options for things like cereal.
I've never had it that bad until these past few months. I usually find it happening when I drink milk or eat something with milk in it. Idk if milk could be the problem.
Some girls just get harder periods than others. Mine were so bad I had to get on birth control to find relief.