Mechanical energy
Mechanical energy is the sum of potential and kinetic energy.
Mechanical energy conservation
Without friction, the sum of an object's initial mechanical energy is equal to the sum of the later mechanical energy.
In other words, the mechanical energy of an object is always preserved.
Consider the case of throwing an object at different angles at the same height.
Potential energy is the same (= Same height).
Kinetic energy is the same (= Same speed).
In other words, there is no difference in the mechanical energy of the throwing angle.
So, regardless of the direction, if the initial velocity is the same, the velocity at the moment of hitting the ground is also the same.