Can we give a definition of the cotangent based on a functional equation?

This might be related. The Herglotz trick is essentially the statement that $\pi\cot(\pi z)$ is the unique meromorphic function $f(z)$ satisfying:

$f(z)$ is defined for $z\in\mathbb{C}\backslash\mathbb{Z}$

$f(z+1)=f(z)$

$f(-z)=f(z)$

$-f(z+\frac{1}{2})=f(z)-2f(2z)$

$\lim_{z\to0}\left(f(z)-\frac{1}{z}\right)=0$