Question

why the streaching of wire is determined by its shear modulus??

Answer 1

Stretching of wire is not determined by its shear modulus. Stretching depends on Young's modulus of elasticity. It's also called Tensile modulus. 

Answer 2

Normally stretching (extension ) along the length of a wire is determined by its length, Young's modulus and the tensile stress applied.

If the wire is thick, then the above is okay.  But, in reality, the wire is thin, so when it stretches, its diameter and cross sectional area decreases, as the total volume is constant.

Hence there i, inherently a shear stress involved perpendicular to the direction of wire.  So  the shear modulus also is important for a wire.

The ratio of lateral strain (in cross section) to the linear strain (longitudinal strain) is calle d the Poission's Ratio.  This is a constant for a material.

Shear strain is dependent on shear modulus for that material.

  Further We can say -- Young's modulus = Y = 2 G ( 1 + σ )
           here G = shear modulus,  σ = Poisson's ratio.

If a characteristic depends on  Young's modulus,  it also depends on Shear modulus.