Analogies between $(\tan, \sec)$ and $(\sinh, \cosh)$

I think everything follows from your first equation. Since the signs of tan, sec go like the signs of sinh, cosh, this equation tells us that the parametric graphs $$ t\in(-\pi/2,\pi/2) \mapsto(\tan t, \sec t) \qquad \qquad t\in\mathbb R \mapsto (\sinh t, \cosh t) $$ consist of the same points in a different parameterization (in fact it's the upper branch of a hyperbola).

So if we define $f(x) = \sinh^{-1}(\tan t)$, then we have $$ \sinh \circ f = \tan \qquad\qquad \cosh \circ f = \sec $$ It's just a particular transformation of the horizontal axis that make the functions into each other.

This means that we have $x\oplus y = f^{-1}(f(x)+f(y))$; in other words $\oplus$ is just ordinary addition transfered through this bijection.

And this also means that your $\mathring D$ is just ordinary differentiation transfered through the bijection too.

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The same $f$ will turn also $\sin$ and $\cos$ into $\tanh$ and $\operatorname{sech}$, for more correspondences.