if mass is assumed to be constant, the velocity of the centre of mass of the system has to be different after the collision for the kinetic energy to be different.
However, if the momentum of the system is conserved, the velocity of the centre of mass of the system should remain the same.
- mass is not constant and velocity is different: in a completely inelastic collision the two objects (A: m =1, B m = 2) stick together and mass becomes A+B = 3
Suppose that $v_a = 6m/s$ and $v_b = 0 \rightarrow E_k = 0.5 * 6^2 = 18, p_a = 1 * 6 = 6$
After collision velocity would be anyway lower as KE should be distributed among more mass, but some KE is lost in the crash. How much?
Momentum is conserved: $ p_{ab} = 6$ , from this datum you can calculate its velocity: $$v_{ab} = \frac{6}{3} = 2$$ and $E_k = 0.5 * 2^2 *3 = 6$, two thirds of energy have been lost
how can there be a change in kinetic energy of the system if there is no change in momentum?
A change of KE without a change of momentum is not only possible but very frequent, because as you noted p varies linearly and KE quadratically. You get the same product by a wide range of factors: 6 = 61 = 32 = 23 = 16 = 0.5*12 etc.
All these are same values for m*v, but as the figure for v must be squared, you get all different values between energy and momentum: KE =3, =6, =9, =18, =72 etc
I hope this clarified all your doubts