Quoting from the linked source: "*thus SWAP has negative
eigenvalues, which means that $T\otimes I$ is not positive and therefore $T$ is not
completely positive*", where $T$ is the transpose. So they are *not* saying that the SWAP is not a realisable operation; they are saying that $T$ isn't.

As you note, the SWAP is a perfectly fine unitary gate. That there are negative eigenvalues is not a problem for a unitary operation (which has in general complex eigenvalues in the unit circle).

The argument made in the linked textbook is that if a quantum map $\Phi$ is such that $\Phi\otimes I$ is not positive, *i.e.* has some negative eigenvalue, then $\Phi$ is not completely positive, and thus does not describe a physical quantum operation.