One can define the notion of "indecomposable" in many of the categories that mathematicians think about. However there's no real reason, as far as I can see, to expect it to behave at all well.
Here are two examples.
1) Say a group is indecomposable if it is not the trivial group, and not isomorphic to a product $H\times K$ with $H$ and $K$ groups, and neither of them the trivial group.
Can one find a group which can be written both as a product of two indecomposable groups, and as a product of three indecomposable groups?
2) Say a topological space is indecomposable if it has more than one element, and is not isomorphic to a product $X\times Y$, with $X$ and $Y$ topological spaces both having more than one element.
Can one find a topological space which can be written as a product of two indecomposable spaces, and also as a product of three indecomposable spaces?
In both cases the notion of indecomposability seems a bit artificial, or at least not commonly used, so I suspect that one can find such funny examples in both cases. I am pretty sure, for example, that it's not hard to write down a number field whose ring of integers contains elements which are both the product of two irreducibles and three irreducibles, and numbers are a lot easier than either groups or topological spaces. I don't know explicit examples for either Q1 or Q2 though. Does anyone else? I was told by an algebraist that no example of a finite group as in Q1 above can exist, which already surprised me a little.
Finite products and finite coproducts coincide in the category of [edit: abelian] groups I guess, but one could also formulate an analogue of Q2 using disjoint unions and this seems much more close to the kind of question that people do actually think about, so we can safely ignore it here :-)