In the question Can one define a “particle” as space-localized object in quantum field theory? it is said that in quantum field theory, a particle state is a state with well defined energy and momentum, related with dispersion relation $E^2=p^2+m^2$. This thing is localized in momentum space, which means it must be delocalized in coordinate space.
On the other hand, in classical mechanics, the most striking feature of a point particle is being localized in coordinate space.
On the first glance, these two objects may seem very different with no obvious link between them. It seems that it is usually just postulated that the above defined QFT states are particles, without any clear justification why should they be related to the point-like classical objects. Satisfying the same dispersion relation isn't good enough justification, as it is not obvious that e.g. some classical field configurations can't satisfy it.
So my question is: Why do we call them both particles, how do we see that they behave similarly (in some appropriate limit)?