Is there a coordinate system where a point is defined with two angles?

Yes. There are infinitely many. An example, for 3d space:

Actually, a coordinate system is no more than a function that maps coordinates to points in a Cartesian coordinate system.

For example, the 2D polar coordinate system is a function which maps the points given by $\begin{bmatrix}r \\ \omega \end{bmatrix}$ to $\begin{bmatrix} r \cdot \cos(\omega) \\ r \cdot \sin(\omega) \end{bmatrix}$.

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In 2D space, you can map 2 angles to the points of the plane by, for example, defining their angle from the point $(-1, 0)$ and from $(1, 0)$:

In this example, we map the points of the 2D real plane by two real angles.

This system has the disadvantage that you can't represent the points of the $x$ axis: all of them would have $0$ or $\pi$ as their coordinates.

In target tracking applications you may have a missile incoming towards your base, and you have two radars providing bearing measurements on the missile. In that case, the missile's location is parameterized by the bearing measurements from two radars.

See my crude diagram where circles are radars and diamond is missile.