I’m a noob so sorry for my ignorance. The question is: do we need (in jet airplane) to increase the throttle to maintain constant thrust output if we want to fly higher (not considering climb phase), I mean whether on 2000 ft and 20000 ft altitude throttle is in the same position or we need to push it forward if we fly at higher altitudes to maintain constant power (as in piston airplanes).
It depends, as so often.
With the thrust available at low altitude, the indicated flight speed will be much higher than at high altitude. There, the reduced density and thrust will only allow flight close to maximum thrust and over a small speed range around the minimum drag polar point. In practical terms, at low altitude the airliner will not be able to fly level at the maximum thrust setting because it will exceed its Mach limit. But since the speed of sound is higher in warmer air, it will still fly significantly faster than at high altitude when it flies at the same Mach number at both altitudes.
If, on the other hand, you compare states at the same indicated air speed, the required thrust at low altitude will be a bit smaller due to the higher Reynolds number in warmer air. On the other hand, fuel flow will be a bit higher since the engine is not running at its optimum design point and the higher ambient temperature reduces its thermal efficiency.
While thrust will go down with air density (or dynamic pressure, for the same Mach number at low and high altitude) in the first case, in the second case it will increase slightly.