This is perhaps a well-known result and I'd appreciate a reference if that's the case. Let $X$, $Y$ be iid random variables with support $S \subset \mathbb{Z}$ and let

$$P(X = x) = f(x, \theta)$$

where $\theta$ represents a vector of a countable number of parameters.

The question is twofold. Firstly, does there exist non-trivial distributions such that

$$P(X + Y = x) = f(x, \theta_1(\theta)), \;\;\; P(XY = x) = f(x, \theta_2(\theta)) ?$$

for some $\theta_1$, $\theta_2$. By trivial here, I mean the case that $f(x, \theta) = \theta_x$, i.e., there is a bijection between $S$ and the set of parameters.

Secondly, within this class of distributions, does there exist distributions with a finite number of parameters?