Is there a precedent for a structure that has two monoids where the additive identity also absorbs with multiplication, but multiplication does not distribute over addition. So it's basically a Semiring except for distributivity. It's close to a Near Ring, but not quite. Is there a name for such a thing?
What you said, plus distributivity on a side, is called a near-semiring.
I doubt the version with no distributivity at all has been named or interested anyone: it is far too loose a coupling between the operations. Without such a condition, there is no interplay between the operations.