Is there a name for sec(x)'s relationship with tan(x)?

I wouldn't say it's "nothing special". In fact it's quite special. If $f(x)$ and $g(x)$ are functions that satisfy $f'(x) = g(x)^2$ and $f(x)^2 + 1 = g(x)^2$, then $f'(x) = f(x)^2 + 1$. But that is a differential equation whose general solution is $f(x) = \tan(x + C)$, where $C$ is an arbitrary constant. And then we get $g(x)^2 = \tan^2(x+C) + 1 = \sec^2(x+C)$, so $g(x) = \pm \sec(x+C)$.

No, there's no name for this relationship - it's a coincidence. Nothing special is going on in the identities you mentioned. In fact, the existence of a separate name "$\sec(x)$" for the function $\frac{1}{\cos(x)}$ is just a historical accident - the identities you stated are really relationships between $\sin$ and $\cos$, or even more properly, really relationships of the function $e^{ix}$ with itself.