No, it does not. It does not change even if the actual wing area does e.g. by extending Fowler flaps.
Lift is only very roughly proportional to wing area. It depends on the span and the chord and the profile in complex ways. By referencing to the wing area you are just getting a number that is of the same order for different wings so you can more easily see how good they are.
Referencing to different area for different angle of attack or different flap positions would just hide part of the effect of those on lift, so you reference to the same area for all flight regimes to have all the effect show in the lift coefficient.
The reference area is arbitrary, i.e. you can choose what your reference area is. But the idea of a reference area is that it is constant, and the influence of e.g. angle of attack on lift is reflected in the coefficient of lift. Some form of projected wing area is just the most common choice for airplanes (and there are certainly some good engineering reasons for choosing it).
If you compare drag coefficients for cars and airplanes, the former are usually based on frontal area, whereas the latter are based on what is roughly the wing area (typically including part of the fuselage), so you can't even compare dragginess between the two types of vehicles by just looking at the drag coefficient.