I'm not sure what you mean by "common mozzarella." If you mean the dried out "low-moisture" stuff you find in the U.S., that's just not common in Italy. If your pizza actually had "mozzarella" on it, it was likely either actual mozzarella di bufala (from buffalo milk, the traditional version) or fior di latte (i.e., cow's milk mozzarella, which we'd call "fresh mozzarella" in the U.S.). Those two are pretty much the definition of "mozzarella" in Italy. Other cheeses or forms would rarely be called simply "mozzarella." Mozzarella di bufala is more expensive and is often advertised specifically on a menu, though some purists might say that just the term mozzarella by itself should indicate buffalo milk (as opposed to fior di latte).
Now, there are commercial forms of fior di latte available in Italy which have lower moisture content specifically for pizzas, though I don't think anywhere near as low-moisture as the typical stuff you find in an American supermarket. Other pizzerias will drain the fresh fior di latte for several hours before using on a pizza to avoid excess moisture and an excessively soggy pizza.
All of that said, Italians put all sorts of different kinds of cheese on pizza. It's not uncommon at all to have a grated hard cheese be the only cheese on a pizza in some regions, or even no cheese at all (e.g., Rome's pizza bianca). It's quite possible you were served pizza with some other "melty" cheese too, but it wouldn't have been called "mozzarella" on a menu.
EDIT: I see now that the OP is from Brazil. I don't know what sorts of mozzarella are common in Brazil, so I don't know what "common mozzarella" means precisely in the question (as I said already). Just for clarification: in the U.S., there is a very dry "low-moisture" form of mozzarella often used for pizzas. My experience is that in other countries, the lower moisture form of mozzarella is still quite moist, though it can be sealed in a plastic wrapper without brine/packing liquid. In the U.S., this form of mozzarella would generally still be considered "fresh mozzarella," just without brine. However, obviously mozzarella packed like this is rarely "fresh" as in Italian "fresh" mozzarella, which is often consumed the day it is made. So again, I think the exact difference between "common mozzarella" and "fresh mozzarella" would vary significantly by country, but hopefully my answer above is still clear.