Why are the formulas for price elasticity of supply and demand the same in intro microeconomics? I get that they are both gauging responsiveness, but I don't particularly understand why they are the same formula.
The elasticity is a very general concept. Consider an arbitrary function

$$y=f(x_1,x_2, \cdots , x_n) $$

The elasticity of $y$ with respect to $x_i$ is

$$ e_{y,x_i} = \frac{\partial y}{\partial x_i}\frac{x_i}{y} $$

It measures the percentage change of $y$ when variable $x_i$ changes by 1% - a measure of responsiveness, as you say. It can be applied to anything, including the demand and the supply, where you get the common concepts like own-price, cross-price, income elasticity, etc. The sign of the elasticity is given by the sign of the derivative, which depends on the function $f(\cdot)$.

Elasticities applied to functions other than standard demand and supply abound. For example, individual factors of production, relative factors of production, investment, taxation, preferences, city size, et cetera.