Gravity isn't just about mass, but about distance, too.
Our moon has a surface gravity of about 1/6th of Earth, because it is small and less dense than the Earth is. Surface gravity of a body is inversely proportional to the square of its radius. That means that if you compressed the moon such that it was $\frac{1}{\sqrt{6}}$th of its current radius, it would have the same surface gravity as the Earth even though its mass woudn't have changed at all.
It would have to have a density of about 50 tonnes per cubic metre though, and that's heavier than any normal material so this situation couldn't arise around Earth. You'd need to arrange for a very dense metallic moon to orbit a small or very low density planet... perhaps one mostly made of ice or water. It would be a little surprising to have this arrangement, but not actually impossible. Just unlikely.
As an example, you could imagine a planet a bit like Callisto, which has a surface gravity of about 1/8th of Earth despite its size due to being made largely of ice and rock. A spherical moon with a radius of 200km made of iridium would have a slightly higher surface gravity, but still weigh less than 1/150th of its parent planet.