Does any given integer only occur in one primitive Pythagorean triple?

Yes, a number can appear as the smallest value in two distinct primitive triples. For example, $(57, 176, 185)$ and $(57, 1624, 1625)$.

In fact, choose any two relatively prime positive integers $p$ and $q$ with $q+1 < p < q(1+\sqrt{2})$ and having opposite parity. Then $p^2-q^2 < 2pq$, and $(a,b,c)=(p^2-q^2, 2pq, p^2+q^2)$ is a primitive triple. Another primitive triple with $a$ as smallest element can be derived from the generators $r=\frac{a+1}{2}, s=\frac{a-1}{2}$: $$(r^2-s^2, 2rs, r^2+s^2) = \left(a,\frac{a^2-1}{2}, \frac{a^2+1}{2}\right).$$

The example $(3,4,5)$ and $(5,12,13)$ shows that some positive integers can appear in more than one primitive triple.

There are integers that are in more than one primitive triple. For example $5$ occurs in $(3,4,5)$ and $(5,12,13)$. The number $65$ occurs in $(33,56,65)$, $(65,72,97)$ and $(63,16,65)$.