In the wiki page of pointed set, they refer to the zero object as:
the pointed singleton set $(\{a\},a)$
It seems that $(\{a\},a)$ and $(\{b\},b)$ when $a \ne b$ are not distinguished. Is it the same case in the category of sets (Set)? That is, is there only one object representing singletons in Set, only one object representing two-element sets, and so on?
No, every set is a distinct object in the category Set.
However, all category-theoretical properties of objects are preserved by isomorphisms. In particular, if some object is a zero object, then any object isomorphic to is also is a zero object. We say that there is a unique zero object, up to isomorphism.
For sets, an isomorphism is just a bijection; any two sets with equal cardinality are isomorphic. So, while there are a lot of one-element sets, they share the same properties from category-theoretic viewpoint, like do all two-element sets, and so on.