Why are hyperbolic functions defined by area?

The unit circle is the locus of points with unit distance from the origin in the Euclidean metric: that is, the metric corresponding to the standard dot product $(x_1,y_1)\cdot (x_2,y_2)=x_1x_2+y_1y_2$. You can define the standard trig functions as parameterizations of the unit circle in the Euclidean metric, using either arc length or area; both definitions will be equivalent.

Similarly, the unit hyperbola is the locus of points with unit distance from the origin in the Minkowski metric: that is, the metric corresponding to the nonstandard dot product $(x_1,y_1) \cdot_M (x_2,y_2)=-x_1x_2+y_1y_2$. Again, you can define the hyperbolic functions as parameterizations of the unit hyperbola in the Minkowski metric, using either arc length or area; both definitions will be equivalent.

So why does it appear that you can define the hyperbolic functions using area, but not using arc length? Because you're studying the hyperbola in the Euclidean metric! The Euclidean arc length of a curve $\displaystyle \int_C \sqrt{dx^2+dy^2}$ is usually different from the Minkowski arc length $\displaystyle \int_C \sqrt{-dx^2+dy^2}$ of that curve, so switching metrics leads to a different arc length parameterization for the hyperbola.

However, both the Euclidean and Minkowski definitions of area turn out to be equivalent. To see this, note that we can just show it for parallelograms, and then integrate to get the result for arbitrary shapes.

The signed area of the parallelogram spanned by the vectors $v_1=\left<x_1,y_1\right>$ and $v_2=\left<x_2,y_2\right>$ can be computed via the dot product as follows. First, let $v_1^\perp=\left<-y_1,x_2\right>$ be one of the two vectors perpendicular to $v_1$ (that is, with $v_1^\perp \cdot v_1=0$), and with the same magnitude as $v_1$. Then the signed area of the parallelogram spanned by $v_1$ and $v_2$ is $x_1y_2-y_1x_2$, which can be conveniently written in the form $v_1^\perp \cdot v_2$.

Working in the Minkowski metric, if $v=\left<x,y\right>$, then the vector perpendicular to $v$ with the same magnitude is $v^\perp=\left<y,x\right>$. So if $v_1=\left<x_1,y_1\right>$ and $v_2=\left<x_2,y_2\right>$, then $v_1^\perp \cdot_M v_2=-y_1x_2+x_1y_2=x_1y_2-y_1x_2$, which is coordinate-wise identical to the expression for the area in the Euclidean metric.

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TL;DR: if we define the circle or hyperbola in the most natural metric for that particular curve, we can get the trig/hyperbolic functions using either arc length or area.

But most of the time, we stick to the Euclidean metric when doing coordinate geometry. The area definition of hyperbolic functions turns out to be the same in either metric, but the arc length definition doesn't, so it ends up looking like we need to use the area definition and not the arc length definition.