Most textbooks explain intensive coordinates by asking us to consider a system and divide it into two parts. The properties which remain the same will be called intensive and the properties that change are called extensive. In other words intensive variables are independent of mass (so what is written in my textbook: "The intensive coordinates of a system, such as temperature and pressure, are independent of mass and the extensive properties are proportional to mass). Now say I have a system of gas of mass m and I introduce more mass m keeping the system's volume constant. Have I not increased the pressure and maybe temperature of the system too in doing so?
If you add mass keeping the volume constant, the pressure would change.
When a source says that pressure is independent of mass, it means that pressure is independent of mass if you keep the density constant, not the volume.
The idea is that if you take two identical copies of a system, then combine them to make a system twice as large, extensive variables will double. That includes mass and volume. Intensive variables like pressure and temperature will stay the same.