The consistency with the underlying abelian group could be restated as "multiplication is bilinear, and has a unit", which is the correct perspective to take if

The consistency with the underlying abelian group could be restated as "multiplication is bilinear, and has a unit", which is the correct perspective to take if you want to "upgrade" from the usual definition to the "monoid object in the category of abelian groups" definition. But understanding bilinearity and its connection to tensor products is the key part of changing between those definitions, and also the most useful step to understand. The final description is pretty uninteresting in contrast with universal mapping properties of a tensor product.