I would like to ask you if this reasoning is correct:
Let $ S $ be the set $ \{1,2,3,4\}$ and let us define a partial ordering $\leq$ for the two pairs $\{1,2\}, \{3,4\}$ as $1 \leq 2$ and $3 \leq 4$. We assume that the other pairs are not comparable. I understand that this means that $\{1,2\}$ and $\{3,4\}$ will be chains with the upper bounds $2$ and $4$. Then, if I understand it correctly, Zorn's lemma states that $S$ will have a maximal element $m$ such that for each $a \in S$ if $a \neq m$ then it is not true that $m\leq a$. It seems to me that this means that there are $2$ maximal elements in this case, one for each chain. Is this reasoning correct?
Yes. From the definition of a maximal element, a given poset can have more than one maximal element. In your case, the maximal elements are 2 and 4.