i expect they can be, but why would anyone bother with them, when the "square" functions can easily be adapted for the purpose?
An angle in radians is defined as the length of the arc of a circle over its radius.
A slight problem occurs if you want to extend that to an ellipse because an ellipse is defined by two variables, major axis and minor axis. It would be interesting how to define an elliptic angle, length of arc of an ellipse over major axis or minor axis or their algebraic combination? But in either case, I don't know how this could be useful other than perhaps in an elliptical coordinate system.
If you google elliptic trig there is lots of different papers on it.