The quoted comment from the paper is parenthetical and follows an assumption about integer polynomial $P(x)$ being a product of factors "irreducible over $\mathbb{Q}$":
$P(x) = h_1(x) \cdot \ldots \cdot h_\nu(x)$
The paper's authors then note their assumption that the GCD of coefficients is 1, which defines that polynomial $P(x)$ is primitive.
The presence of a nonunit common divisor of coefficients would make the product as irreducibles over $\mathbb{Q}$ "non-unique". Before describing that implication, note the authors own remark, still within the parenthetical comment, that "this has no bearing on the paper, since for our results $P$ may be replaced by" $P$ divided by the GCD of its coefficients, which would then be an equivalent (for the sake of finding roots) primitive polynomial.
Uniqueness of a factorization always has two conventional caveats: order of factors is not important (when multiplication is commutative) and the factors themselves are only unique up to multiplication by a unit (associates). Therefore moving a nonunit constant divisor from one factor to another creates a different factorization. It also is not allowed to treat the nonunit constant divisor as one of those factors "irreducible over $\mathbb{Q}$", because the constant divisor becomes a unit as a rational number.
It might have been clearer if the authors had stated that primitivity of $P(x)$ could be assumed without loss of generality (by dividing out any nonunit constant divisor), and not raising unnecessarily an issue of non-unique factorizations.