It’s also the standard notation for zero objects, i.e., for terminal objects that are also initial. This entails all abelian/additive/preadditive categories, such as categories of modules, vector spaces, or abelian groups. (But there are also counterexamples, such as the categories of groups and of pointed sets.)
But I’d agree that it’s not standard notation to use 0 for a terminal object in an arbitrary category. I’d guess that most people use 1 instead, so that for example 1 × X ≅ X. (The post talks about group objects in the category of algebraic varieties (over some field), in which case 1 seems to be more appropriate than 0.)
