Welcome to HCM,
As you have undergone tonsillectomy
2 days back and you are getting itchiness and cough
after drinking milk. This could happen following the tonsillectomy as your tonsils are removed there will be irritation in the tonsillar bed which may produces these symptoms.
It can happen for first few days til the wound is healed, any fluid or food you take for initial few days these symptoms can persist. Nothing to worry about this.
For your symptoms I would suggest you to follow
1. Eat Ice cream
like Tab BActcoclav.