Is the product of two primes only factorisable by those two primes?
Yes, that is true (unless you count the trivial factors of 391, namely 1 and 391).  The fundamental theorem of arithmetic states that any postive integer is a product of a unique multiset of primes, and that no other multiset will multiply together to form that integer.

And so, since $391$'s multiset of primes is $(17, 23)$ (because both $17$ and $23$ are prime, and $17 \times 23 = 391$), that implies that no other prime can possibly divide $391$.  If there were such a prime $p$, then we'd have an alternative multiset that consisted of $p$ and a multiset that was the factorization of $391/p$; that would be a different multiset (because, by assumption, $p$ is neither $17$ nor $23$) with the same product, and we know that can't happen.