As another commenter said, a square is a unit of the plane; a square figure; all sides equal. Likewise a radian is a unit of the plane; a pie-slice figure with all sides equal.
Pie-wedge-shaped things are not the only use for radians.
The phase [1] across a Josephson Junction is measured in radians, but there are no planes involved. The progress of a longitudinal wave [2] as seen passing a stationary observer is measured in radians per second, yet no plane is involved -- it works just fine in a one-dimensional universe.
Radians measure cycles. If you limit yourself to geometry then sure, radians seem like a thing for measuring circles. But that's just a side effect of the fact that the only cyclic symmetries of Euclidean space are rotations. So the only cycles you're going to find in high school geometry will be related to circles.
