I know that the circumference of a circle of radius $R$ in hyperbolic geometry is $2π sinh R$. I believe this is approximately equal to $πe^R$.
I know that the area of a circle of radius $R$ in hyperbolic geometry is $2π(cosh R − 1)$. But this also appears to be approximately equal to $πe^R$.
I plotted all three of these functions on Wolframalpha, and the graphs do appear to be relatively identical when the range on the x-axis is large.
Have I made some mistake here? How can the circumference be equal to the area?
They aren't equal, but as you've noticed they're approximately equal.
What's likely happening is that you've discovered that: $$\sinh(x) = \frac{e^x-e^{-x}}{2},\qquad \cosh(x) = \frac{e^x+e^{-x}}{2}$$
So, we have that: $$2\pi\sinh R = \pi e^R-\pi e^{-R}$$ For large $R$ we get that $e^{-R}$ gets small, so we will get that $2\pi\sinh R\approx \pi e^R$.
Now, we have that: $$2\pi(\cosh R-1) = \pi e^R+\pi e^{-R}-2\pi$$ There's no reason that this should be approximately equal to $\pi e^R$, unless you're choosing $R$ large enough that $\pi(e^{-R}-2\pi)$ is negligable compared to $\pi e^R$. In this case, we would again have that it's approximately $\pi e^R$.
In both cases, they're aren't exactly equal. They can be equal to good approximation for large $R$ though.