While this product of additive inverses property does follow from the field axioms, there is a short discussion in the blog comments at https://susam.in/blog/product-of-negatives/comments/ which shows that this property holds true for all rings too.

So it is not just numbers for which this property holds true but for all elements of fields and rings too. Quite simply, (-a)(-b) = (a)(b) in all rings where (-a) and (-b) are the additive inverses of a and b respectively.