Paper airplanes' wings are completely flat, unlike the droplet shaped curvature of a full sized wing. How does a flat wing generate any lift if both sides have the same air pressure?
Paper airplanes create lift just like any other airplane. Don't let the wing cross section confuse you; what the air "sees" is a different shape.
When a flat plate flows through air at a positive angle of attack, the stagnation point of the flow (where air splits into an upper and a lower flow path) sits slightly below the forward edge of the wing. Now the streamline just above the one which hits the stagnation point has to negotiate the sharp corner of the leading edge, which will cause it to separate from the surface. This creates a small separation bubble which to the streamlines further up looks like a round leading edge, and since the rest of the wing has no curvature, the flow will attach again soon. The bigger the angle of attack gets, the farther back the stagnation point will be and the bigger the separation bubble becomes. At some point, which is promoted by leading edge sweep, the separated flow will not reattach, but produce a vortex, like the vortex which provides lift on delta wings.
The picture above is from a simulation; see the full video here. Note the red color on the forward part of the upper side: This indicates low local pressure, which is needed to make the outer air negotiate the leading edge plus the separation bubble (see here for an explanation on the molecular level). In the separation bubble, low pressure is not accompanied by high flow speed since the separation turbulence has eaten up most of the kinetic energy.
The actual nose radius of the paper wing does not matter, and it might even be zero (like when you fit razor blades to the leading edge): The air will create its own airfoil shape by padding with separated air what cannot be reached with attached flow.
Paper airplanes have a low enough wing loading that their angle of attack is low and the separation does not become too big. All it needs is a center of gravity at the quarter chord of the wing; that is why you need to fold the piece of paper first.
It is interesting to watch a slightly tail-heavy, delta-shaped paper airplane fly: It will gradually reduce speed and will start to rock its wings when it approaches stall. Once one wing is going down, the sudden increase in its local AoA will increase lift so much that the rolling motion reverses, without noticeable yawing. The delta-shaped flat plate has effective roll damping right into the stall. The lack of yaw is mostly due to the high yaw inertia, but indicates also that drag buildup is moderate at best.
Of course, after a few periods of rocking the nose will drop and it will pick up speed like any well-behaved airplane.
How does a flat wing generate any lift if both sides have the same air pressure?
The two sides do not have the same pressure.
A flat plane can still produce lift (think of putting a plate out of the window of a moving car, if you point it slightly upwards it will strongly push upwards, that's lift). The main reason is the Kutta condition that must be satisfied at the trailing edge: this will create circulation around the flat plate, generate a pressure difference and thus generate lift.
Paper airplanes will inevitably come down not because they do not produce lift, but because they also have drag (and no thrust), that will slow them down and hence reduce the lift generated.
It's funny how many people misunderstand such a fundamental principle of flight like how an aircraft wing creates lift.
Most misunderstandings stem from incorrect textbooks written by unqualified people who know about flying, but not about physics. This incorrect information is often passed on at flight training schools by flight instructors who were themselves given wrong information.
Your question demonstrates this point exactly. I could reword the question to state: "If a paper aeroplane flies with straight wings, does this mean that the shape of an aircraft wing is not the thing that actually generates lift?". The answer is self-evident.
The incorrect analysis of lift usually goes something like this: an aircraft wing creates lift because the air travelling over the curved surface on the top must travel faster than the air moving along the shorter surface underneath. In accordance with the Benouli Principle, the faster moving air creates a low pressure area that sucks the wings upwards.
How then, does an aircraft fly up-side-down? Wouldn't the low pressure area now be underneath, pulling the aircraft towards the ground? How does a paper aeroplane fly with a flat wing? The answer is that it is not the shape of the aircraft wing that generates the lift.
An aircraft wing generates lift due to the physical reactions of the wing surface coming into contact with the air as the wing moves through the air, deflecting air down. In accordance with Newton's Third Law of Motion, that downward force exerts an equal and opposite upward force on the wing, creating lift. Increase the angle of deflection caused by the wing (known as the Angle of Attack) and the force generated increases, creating even more lift at a given airspeed, at least until it reaches the critical Angle of Attack, where the slowing force created by the increased drag eventually exceeds the lifting force, causing the aircraft to stall.
So why aren't aircraft wings flat? As in nature (in the case of birds), the curved surface of an aircraft wing helps to reduce drag, allowing air to flow smoothly over the wing without separating and becoming turbulent. Lower drag means a more efficient wing.
If you want to know more, NASA has some good information on this topic on their website. You can check it out here: https://www.grc.nasa.gov/www/k-12/airplane/lift1.html
Because it takes the path of least resistance. There is lot less drag in slipping forward through the airmass as it falls than in falling flat towards the earth.
Think about it: If you stuck a putty knife in a block of jello at a 45 degree angle, then pulled it straight towards you, would it come straight towards you, or might the blade tend to angle towards one side or the other? (depending on which direction the closer edge was...)
Would you say that this phenomenon is due to a differential in gelatinous pressure, or that the putty knife produced lift? I wouldn't. I'm not an engineer, but I would say that the knife simply took the path of least resistance.
It is true that a flat wing can produce lift, but a paper airplane's wings are not flat on top. They have a folded piece on top that bows slightly up. Air has to flow a longer distance over that slight curve as compared to the bottom of the wing, just like an airplane. The faster air causes a pressure drop relative to the bottom of the wing, thus lift.