Is projectile motion an approximation?

Just as the motion of body around the earth is ellipse (1st Kepler law replacing sun by earth), so is the motion of a projectile. Notice that almost everything we deal is an approximation, the earth is not a massive perfectly rounded ball, and we are neglecting the air, so it is not a sin to consider the motion of the projectile as a parabola (at least in our everyday experience as, for example, throw a stone in the river)

Assume Galilean relativity and Newtonian gravity. Neglect the drag due to the atmosphere. The gravitational field of the Earth is the same as the one produced by a point particle in its center (with the same mass, the usual $1/r^2$ gravitational force field). Now, you may know that a test particle in this $1/r^2$ force field of the Earth can have different orbits (closed or open, depending on the initial velocity and the initial position). Leave out open orbits, which means that you are shooting the projectile at infinity. All other orbits are ellipses. However, the Earth is not a point and has a finite radius: some of those ellipses (starting at the Earth surface) will intersect at later times the Earth surface again.

Why do you have parabolas in the "simple" setting you are describing? Because you can always approximate locally an ellipse with a parabola. See e.g. Can a very small portion of an ellipse be a parabola?

The other answers have already mentioned the approximations related to gravity, but there is more to that when discussing a projectile motion:

• For example, one typically neglects friction, and if one were to include friction, the choice of the formula for the friction term would be an approximation for a particular situation (proportional to velocity or its square or cube), and the friction coefficient will be an approximation, since it actually depends on the shape of the projectile.
• One could then also take into account that the projectile has finite size, and that it may rotate in 3 dimensions while flying.

One could go on further, as much as one's knowledge of the situation allows. The art of doing physics is to a large extent one's ability to develop models closer or less approximating real situations, capturing the most essential details, and solving them mathematically.