Can we use the volume to fix the thermodynamic state of a system?
If not, why do we use P-V curves to know about thermodynamic states and processes?
As Chester Miller pointed out, volume alone does not fix the thermodynamic state of a system. The reason we use PV curves for closed systems is for determining total work. Insofar as using specific volume $v$ or total volume $V$ is concerned, since mass in a closed system is constant (not a variable) you normally use total volume $V$ to find total work done. For open systems where mass and/or mass flow rates may be variables, you would use specific volume in computing work per unit mass. For example, for an open system with steady mass flow rate we can obtain the rate of work (power) by multiplying the work per unit mass (kJ/kg) times the mass flow rate (kg/s).
Hope this helps.