Imagine that you have 2 objects colliding without external forces. Then the total momentum has to be conserved, let this be p_tot. Then

p_tot = p_1 + p_2

where p_1 and p_2 are vectors and are the momenta of the objects after the collision. There is an infinite number of choices of p_1 and p_2 that satisfies this relation. If you're not immediately convinced by this, just consider picking p_1 first. You can choose from all possible vectors, and when p_1 is chosen p_2 is determined by the equation above.

When you also require an elastic or inelastic collision this is the constraint that forces p_1 onto one of these vectors, thus also determining p_2. So the momenta that satisfy an elastic collision are not the same ones that satisfy an inelastic one.