Why is amorphous classified solid? Because it does not have a crystal structure, it is hard to find physical similarities with a solid. Why isn't it then another state other than solid?
The physics of amorphous is also quite different from crystal solid.
 A: Glass is a typical amorphous solid.
Amorphous materials typically show no melting point but do have a Glass Transition Point ($T_g$). Below it, the material behaves like a solid, with a glass-like fracture surface when fractured.
Typical amorphous materials include several types of elastomer (rubber) like natural rubber (NR), with a $T_g$ of around $-80\:\mathrm{Celsius}$. Above that temperature NR really behaves like a super viscous liquid, creating the illusion of solidity. This explains of course why NR above its $T_g$ doesn't behave mechanically like a crystalline substance.
When cooled to below their $T_g$ these materials become brittle like glass (while remaining amorphous), see the famous demonstration of the brittleness of a rubber hose when cooled in liquid nitrogen.
A: 
"Because it does not have a crystal structure, it is hard to find physical similarities with a solid" 

simply proves that you're using the wrong definition of "solid". Relatively few solids have crystalline structure, so a good definition of solid simply can't depend on that.
Let me see if I can offer a better definition.
Rigid body (Idealization) The parts of a rigid body maintain exactly the same distances between one another over time. 
Solid The parts of a solid obey an approximate version of that rule for rigid bodies (subject to deformation under stress). The atoms and molecules are not free to move past one another but remain near a set of fixed equilibrium positions in some body-fixed coordinate system, thereby maintaining approximately the same distance between one-another over time.
Under that rule glass-like materials are obviously solids and obviously fluids at higher temperatures. Unlike normal solids, however the transition between states happens gradually aver a range of temperatures.
