I can't give a substantive answer to this question, but differential graded abelian monoids have been sighted in the wild here:
A Chain Functor For Bordism Author(s): Stanley O. Kochman Source: Transactions of the American Mathematical Society, Vol. 239 (May, 1978), pp. 167-196 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1997852
