To clarify why normal reaction is equal to weight you can alwas think graphically. Equilibrium equations are anlytical notation of vector forces. If you draw free body diagram of a body in equilibrium it's always much easier to understand problem and set equilibrium equations. Drawing free body diagram you draw force vectors or better say draw graphical representation of acting forces.
Here is free body diagram of your problem, normal reaction forces act in pairs. You can analyze each body individually or use solidification principle. In case where person's feet do not touch ground:
If you now set equilibrium equations you could calculate each normal reaction and in your case prove that normal reaction or as you said force with which chair acts on you is same as your weight (in case when your feet do not touch the ground).
Also you could use solidification and analyze human and chair as one body. And have only 2 normal reaction (if feet are not touching ground).
If you think in view of gravitational field it may seem to you that this normal force of chair is invented and you may be right but also you cannot dismiss that chair is between body and earth and gravitational fiel tends to bring body cleser by deforming this chair. The rigid matter of chair or better say connections between particles oppose this tendency in the same ammount as the weight of body. The weight of chair is very small in comparison to earth and it does not significaly affect mass of Earth.