Differential probe design ground copper pour

The designer probably did that to reduce the capacitance of the probe.

If you filled that area with ground, when you contacted the probe points to your circuit under test, you'd be connecting a capacitance between the probed points, and that could change the behavior of the circuit.

Even without the ground, you're adding some capacitance, but removing the ground minimizes the effect.

What would happen if i pour ground copper at that area as well.

The PCB legend implies it's good for 1 GHz. At 1 GHz a 1 pF capacitor has an impedance of 159 ohms. Do you really want to spoil the lack of capacitance at the probing end of the circuit with extra capacitance to ground circa +100 ohms on each input line (because of extending the ground plane)? It may not make that much difference when measuring many circuits but there's no need to do it given the input is differential (by the look of it).