Strictly speaking, a definition doesn’t need justification. However, some definitions are useful and important, and it is relatively easy to show why they are useful. In the case of momentum, basically all of the experimental observations that confirm Newton’s third law justify using $\vec p=m\vec v$ as an important quantity.
If Newton’s 3rd law holds then for any isolated mechanical system the quantity $m \vec v$ is constant, or conserved, regardless of the details. This is a very useful property and so it makes sense to give it a specific name and traditional symbol.
Observations supporting Newton’s 3rd law include (in no particular order):
Recoil of a gun