This is really two questions.
Why is the Lambert W function alternatively called the product log? I have not found any reference to why it is called that, only that it occasionally is.
On the Wikipedia page for the Lambert W function, it states that the function has two branches, $W_0$ and $W_{-1}$. But on a paper on the Wright Omega function, the authors state that there are infinite branches:
Lambert W satisfies $W(z)exp(W(z))=z$, and has an infinite number of branches, denoted $W_k(z)$, for $k\in \Bbb Z$.
Which is correct?
The Lambert W is the inverse of
$$f(z) = ze^z$$
Since the inverse of $f(z) = e^z$ is called logarithm, it makes sense to call the inverse of the product $ze^z$ product logarithm.