[Figures would really help this answer. I'll see if I can add some later.]
According to one definition, the circular and hyperbolic functions parameterize the circle and hyperbola (respectively) according to an appropriate definition of "radian" measure of an angle.
Let $\angle IOP$ have its vertex at the origin, point $I$ at $(1,0)$, and point $P$ somewhere above the $x$-axis. For convenience, let $P$ lie on the unit circle ($x^2+y^2=1$), and let ray $OP$ determine the point $P^\prime$ on the "unit hyperbola" ($x^2-y^2=1$). Then the (circular) radian measure of $\angle IOP$ is twice the area of circular sector $IOP$; and the hyperbolic radian measure is twice the area of the hyperbolic sector $IOP^\prime$ (provided that the angle is no greater than half a right angle). Thus,
We define the circular trig functions by calling $T(\cos(t), \sin(t))$ the point on the unit circle such that $\angle IOT$ has (circular) radian measure $t$.
We define the hyperbolic trig functions by calling $T\,^\prime(\cosh(t), \sinh(t))$ the point on the unit hyperbola such that $\angle IOT\,^\prime$ has hyperbolic radian measure $t$.
(It is a worthwhile exercise to show that the hyperbolic functions defined in this way have standard explicit representations in terms of $t$. Here's a sketch: Rotating the hyperbola $45$ degrees and scaling appropriately yields the curve $y=1/x$. From there, the definition of the natural logarithm allows for determining the area of the hyperbolic sector from $T\,^\prime$, so that the exponential appears when reversing the process.)
It is not at all unreasonable to seek "parabolic" and "elliptical" (and, let's not forget, "not-rectangularly-hyperbolic"!) variants of these trig functions. (In fact, I made the very same investigation as a high school student.) The key is to decide upon exactly what curve to use, and how to assign a "generalized radian measure" to a given angle. I'll describe an approach that generalizes circular trig functions to conics of every eccentricity, $e$; the downside is that it doesn't match the standard hyperbolic trig functions for $e=\sqrt{2}$.
We simply generalize the centered unit circle to a conic with focus at the origin and with "semi-latus rectum" length $1$. More specifically, letting the "other" focus drift down the negative $x$-axis as $e$ approaches $1$ and drift in from the positive $x$-axis as $e$ exceeds $1$; that is, we'll concern ourselves with the (branch of) the conic with closest vertex to the right of the origin, "opening" to the left. Such a conic has equation
$$x^2( 1 - e^2) + 2 e x + y^2 = 1$$
which is the template for the generalized "Pythagorean relation" of our new functions.
The vertex, $V$, of (the preferred branch of) our conic has $x$-coordinate $v := \frac{1}{1+e}$. Let $P(p,q)$ be a point on the curve, with $q\ge 0$. Then we define the generalized radian measure ("grm") of $\angle VOP$ as twice the area of sector $VOP$. The area of the sector is equal to the full area under the curve from $P$ to $V$, adjusted by the area ($\frac{1}{2}|p|q$) of the right triangle with hypotenuse $VP$ (subtracting the triangle if $p<0$ and adding otherwise, so it turns out that removing the absolute value sign from $p$ will "do the right thing"). That is,
$$\begin{eqnarray*}
\mathrm{grm}(\angle VOP) &:=& 2 \left( \frac{1}{2} p q + \int_{p}^{v} \sqrt{1-x^2(1-e^2)-2 e x} \; dx\right) \\\\
&=& p \sqrt{1-p^2(1-e^2)-2ep} + 2\int_{p}^{v} \sqrt{1-x^2(1-e^2)-2 e x} \; dx
\end{eqnarray*}$$
(This explains the disconnect with the standard hyperbolic trig functions. We're defining our radians from "convex" sectors based on a focus, whereas the standard hyperbolic functions define them from "concave" sectors based on the center.)
And now, with vigorously-waving hands ... Writing "$t$" for "$\mathrm{grm}(\angle VOP)$", the above gives a formula for $t$ in terms of $p$. Inverting gives, in terms of $t$, a formula for $p$ ... which we interpret as "the generalized cosine of $t$", or "$\cos_e t $". We solve for the "generalized sine of $t$" from the generalized Pythagorean relation
$$( 1 - e^2 )\cos_e^2 t + 2 e \cos_e t + \sin_e^2 t = 1$$
thereby defining $P(\cos_e t,\sin_e t)$ as the point on our generalized conic such that $VOP$ has generalized radian measure $t$.
General(ised)ly speaking, of course.
Details (and roadblocks) are, as they say, "left to the reader". (I suggest tackling the case $e=1$ first.)