This was inspired from an exercise in Lawvere & Schanuel's Conceptual Mathematics.
It asks what objects in $\mathbf{Set}$ (finite sets), $\mathbf{S^\circlearrowleft}$ (endomaps in $\mathbf{Set}$), and $\mathbf{S^{\downarrow\downarrow}}$ (irreflexive graphs) are isomorphic to the product with themselves. I think the answers in these categories aren't too interesting (unless I missed some, let me know!).
In $\mathbf{Set}$ they are $\mathbf{0}$ and $\mathbf{1}$
In $\mathbf{S^\circlearrowleft}$ they are again the empty set, and the one element set (loop)
In $\mathbf{S^{\downarrow\downarrow}}$ they are the empty graph, the "naked" dot (no arrows), and the loop
So, are there any interesting examples that arise, or are they all pretty trivial?
I cannot answer the question whether the concept of an object isomorphic to the product with itself deserves to be regarded as interesting. However, there are "universal" examples.
1) Terminal objects.
A terminal object $T$ is one such that for each object $X$ there exists exactly one morphism $X \to T$. Examples are the one point set in $\mathbf{Set}$, the trivial group in the category of groups, ...
All terminal objects are isomorphic, and it is easy to see that $T \times T$ is again a terminal object (use the categorical definition of products).
2) In the category $\mathbf{SET}$ of all sets: Each infinite set.
This is a special feature of $\mathbf{SET}$. For example, for abelian group it is not true.
3) Infinite products.
Let $X$ be a any object and $A$ be an infinite set. Then $P = \prod_{\alpha \in A} X_\alpha$ with $X_\alpha = X$ has the property $P \times P \approx P$. This comes from fact that $A \times \{1,2\} \approx A$ in $\mathbf{SET}$.