Let $$(p,q)$$ be a pair of coprime integers; the torus knot $$K(p,q)$$ is the unique (up to isotopy) curve on the boundary of a solid torus which is homologous to $$p\cdot \lambda + q \cdot \mu$$, where $$\lambda, \mu$$ are standard longitude and meridian of the solid torus. One can define torus knots even when $$p$$ and $$q$$ are not coprime, and one gets $$n=\text{gcd}(p,q)$$ copies of the torus knot $$K(p/n, q/n)$$; these links are called torus links. Let us suppose that $$n=2$$, so our torus link has two components. I would like to say that the linking number of these two components is non zero. An example of such a link is the Hopf link, which has linking number 1.

Indeed, I've found a formula (https://discover.wooster.edu/jbowen/files/2013/10/Total-linking-numbers-of-torus-links-and-Klein-links.pdf, Theorem 4.2) for the total linking number of torus links, which in the case of two components is the linking number of the components (which is what I need). The formula is

$$lk(T(m,n))=m \big(n- \frac{n}{gcd(m,n)}\big).$$

Ok, great. But for the Hopf link, which is a $$T(2,2)$$ torus link, the formula gives

$$lk(T(2,2))=2 \big(2-1)=2,$$

but the linking number of the Hopf link is 1! Am I missing something obvious?

P.S. If you know a simple way to say that the linking number of two-components torus links is nonzero, which does not involve any general formula, for me it's okay.

I believe the paper you referenced has a typo in Theorem 4.2. It should say that the total linking number $$\operatorname{lk}(T(m,n))$$ of the torus link $$T(m,n)$$ is $$\operatorname{lk}(T(m,n)) = \frac{m}{2}\left(n - \frac{n}{\gcd(m,n)}\right).$$
The formula they give in Theorem 4.2 (without the extra $$1/2$$ factor) does not agree with their own computation in Figure 9. They compute $$\operatorname{lk}(T(6,4)) = 6$$ in Figure 9, but their own formula yields $$\operatorname{lk}(T(6,4)) = 6\left(4-\frac{4}{\gcd(6,4)}\right) = 12.$$
If you work through their proof, you'll see that they accurately count that the total number of crossings between different components of the $$T(m,n)$$ torus link is $$m\left(n-\frac{n}{\gcd{m,n}}\right)$$. Since all of those crossings are positive, it follows that the linking number is $$\operatorname{lk}(T(m,n)) = \frac{m}{2}\left(n - \frac{n}{\gcd(m,n)}\right).$$