How is volume a State function?
As described by the definition of a state function, that it is
  independent of the path adopted to bring about a change in it [...]


This is not the definition of a state function, since it is missing the crucial first part of it. Here is the definition: a state function is a property of a system that depends only on the current equilibrium state of the system, not on the way in which the system acquired that state.

The point is that something is a state function if you can tell its value by knowing just the equilibrium state of a system and nothing else. As Chester Miller said in his comment: tell me the temperature and pressure (i.e. specify the state of a system), and I'll tell you the volume. It is irrelevant how the system acquired that equilibrium state, the volume is determined by just the state.


  Wouldn't volume change differently if we brought about a change in it
  by say increasing/decreasing temperature rather than say pressure?


In general, of course, when changing the system differently, its volume will also change differently. But, if by doing different changes you eventually will reach the same equilibrium state, all the state functions (and not just volume) will have the same values.