Yes Mozzarella Cheese does have lactose in it but the lactose contained in mozzarella cheese is pretty low.
The amount of lactose contained in Mozzarella Cheese is only 1 gram per ounce of Mozzarella Cheese.
So if you're lactose intolerant you can usually still eat small amounts of the Mozzarella Cheese without it causing any stomach pain or issues with your lactose intolerance.
Other cheeses besides Mozzarella Cheese that contain 1 gram of lactose per ounce of the cheese are the cheeses such as provolone, cheddar, Swiss and blue cheeses.
The lactose contained in those cheeses are less than what the lactose amount is contained in milk.
Some people who are lactose intolerant cannot drink milk but have no problems eating some cheeses as well as ice cream in moderation.