At least when the coalgebra $C$ is finite dimensional over a field and the comodules are finite then the cotensor product is equivalently described as taking linear duals, tensoring them over the dual algebra $C^{\vee}$ then taking duals again.This is why I thought it may be an op symmetric monoidal structure.
In any case a commutative algebra can be reconstructed from its symmetric monoidal category of modules I would like to have a version of this for the category of comodules over a cocommutative coalgebra + some additional structure (which is think ougt to be the cotensor product).