How are the Jacobi Theta Functions analogous to the Exponential Function?

The statement "Theta Functions are elliptic analogues of the exponential function" is not correct because it does not specify in which sense they are elliptic analogues. It is more accurate to state that the Jacobi elliptic functions (which are quotients of theta functions) are the elliptic analogues of trigonometric functions in the following sense. The Wikipedia article Jacobi elliptic functions states

The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn for sin.

The Jacobi elliptic functions are doubly periodic which generalizes the single periodicity of the trigonometric functions which they reduce to. In fact, the current notation, along with the notation used by Jacobi is as follows: $$\textrm{sn}(u, k) := \sin( \textrm{am}(u, k)). $$ $$\textrm{cn}(u, k) := \cos( \textrm{am}(u, k)). $$ $$\textrm{dn}(u, k) := \Delta( \textrm{am}(u, k)). $$ When $\, k=0, \,$ then $\, \textrm{am}(u, 0) = u \,$ and this leads to $$ \textrm{sn}(u, 0) = \sin(u), \, \textrm{cn}(u, 0) = \cos(u), \, \textrm{dn}(u,0) = 1. $$ When $\, k=1, \,$ then $\,\textrm{am}(u,1) = \textrm{gd}(u),\,$ and this leads to $$ \textrm{sn}(u, 1) = \tanh(u), \, \textrm{cn}(u, 1) = \textrm{dn}(u, 1) = \textrm{sech}(u). $$ Thus, the Jacobi elliptic functions are a common generalization of the circular and hyperbolic trigonometric functions.

This probably isn't a complete answer, because I'm no expert on $\vartheta$ functions, but from what I do know, they satisfy a couple of properties that are kind of reminiscent of the periodicity of exponential functions.

There seem to be a lot of $\vartheta$ functions on that page, but the definition I know is that $$\vartheta(z, \tau) = \sum_{n=- \infty}^{\infty}e^{\pi i n^2 \tau + 2 \pi i n z}$$

So these gadgets are defined in terms of ordinary exponential functions, and have some periodicity properties.

We have

$$\vartheta(z+1, \tau) = \vartheta(z, \tau)$$ since increasing $z$ by $1$ just spits out a $e^{2\pi i} = 1$ in every summand, and we also have for integer $\alpha, \beta$

$$ \vartheta(z + \alpha + \beta \tau, \tau ) = \vartheta(z, \tau) e^{- \pi i \beta^2 \tau - 2 \pi i \beta z} $$

which is a sort-of quasi-periodicity.