There is a generally very helpful website which says the following about the hyperbolic angle: "...hyperbolic trigonometric functions take area as their argument (called "the hyperbolic angle," but this is just a name and has nothing to do with angles)..."
From studying other sites it would seem that it has a lot to do with angles, in fact it is an angle which is also twice the area under a line from the origin to a point on the hyperbola, aka sector area. Just checking I understand correctly.