Does the tensor product of dg algebras have a universal property? I have not seen anything about this in the literature.
If $C$ is any symmetric monoidal category, then the tensor product of commutative algebras in $C$ is their coproduct. The tensor product of algebras in $C$, not necessarily commutative, is the "commutative coproduct": We have to add to the universal property that the test morphisms commute with each other. Now apply this to $C=$ chain complexes.