You state that "it is easy to imagine a universe where gravity works differently here vs. there". The laws of gravity may be the same everywhere, but how fast we fall down works differently here vs. there.

That's why we made the conscious decision to make the laws of gravity describe just the parts of "falling down" that are regular, along with stating which features of a given situation make "falling down" work differently, and how: acceleration due to gravity is not the same from place to place, but if you have objects A and B, the ratio between the acceleration of A and the mass of B will always be the same (and vice versa). So we use a gravity equation that describes that part, because it's more convenient to have a single equation that describes the regular part instead of a separate equation for every pair of objects.

You state that "it is easy to imagine a universe where gravity works differently here vs. there". The laws of gravity may be the same everywhere, but "how fast we fall down" does work differently here vs. there.

That's why we made the conscious decision to make the laws of gravity describe just the parts of "falling down" that are regular, along with stating which features of a given situation make "falling down" work differently, and how: acceleration due to gravity is not the same from place to place, but if you have objects A and B, the ratio between the acceleration of A and the mass of B divided by the square of the distance will always be the same (and vice versa). The gravitational constant describes that ratio, and the rest of the equation describes how the rest differs from place to place.