Here is a statement of axiom of choice given in Folland's book Real Analysis.
If $\{X_\alpha\}$ is a nonempty collection of nonempty sets, then, $\Pi X_\alpha$ is empty.
What does it mean to say "a nonempty collection"? Isn't it just enough to say that a collection of nonempty sets?
On
You are correct to argue that if we take the empty collection, then $\prod_{\alpha\in\varnothing}X_\alpha$ is in fact $\{\varnothing\}$.
However, this is somewhat confusing, especially when you're coming to this not from a set theoretic point of view. And you want to avoid thinking about the empty product. Since there is only one collection which is empty (regardless to what you claim its elements to be), this is not any actual hindrance to anything practical when stating the axiom of choice.
This can also be attributed to the author not being very careful to examine the edge cases, and just chucking them aside altogether, which can often happen. Especially when the book is on a different topic (in this case, analysis and not set theory).
On
What does it mean to say "a nonempty collection"?
It means that the collection $\{X_\alpha\}_{\alpha \in A}$ itself is non-empty, i.e., that the indexing set $A$ is non-empty.
Isn't it just enough to say that a collection of nonempty sets?
No, because $\emptyset$ is a collection of non-empty sets.
No, they are different. You could have an empty collection of well anything. Whether that is required is another matter that others have answered.
My answer corrected thanks to Asaf's comment.