Multiplication of the reaction and Equilibrium constant $$\begin{align}\ce{2A + 2B &amp;-&gt; 2C }\\ \ce{A + B &amp;-&gt; C}\end{align}$$

Why does the equilibrium constant change?

And why the rate of the first reaction square the rate of the second? I understand it mathematically, but logically and scientifically I can't understand it at all.
Rate will not be square in most of the cases and who said so? The equilibrium constant scientifically changes because Gibb's free energy is extensive quantity and it is related to equilibrium constant through relation that you might know $\Delta G=\Delta G^\circ +\mathcal{R}T\ln Q $. Where $Q$ is reaction quotient which at equilibrium becomes equilibrium constant and then it gives $\Delta G^\circ = -\mathcal{R}T\ln K$