Mars doesn’t have ‘moons’ - it has 2 captured asteroids, one of which, Phobos, is likely to break up and form a ring, or rings, Soon™ (about 50 million years).
Mars’ ‘moons’ are tiny - Deimos is 12 km wide and Phobos is 22 km wide. Because these are the only natural structures in orbit around Mars, it is unlikely that they formed around Mars, which is why they are generally accepted to be asteroid captures.
Earth is more a binary system of one large and one small planet-sized bodies - current accepted theory is that Earth was sideswiped by a Mars-sized planet (Theia) very soon after it formed, and the result was Earth as we know it and the 5th largest moon in the whole Solar System, (after Ganymede, Titan, Callisto, and Io - it is larger than Europa and all other solar system moons).
The ratio of Earth-Moon sizes is only exceeded by Pluto-Charon, even though our Moon is actually larger than Pluto (about 1.6 times larger).
The Earth-Moon system has large enough Hill Spheres to sweep most detritus from their orbit, the only uncleared materiel being small rocks in the trojan points of the Moon’s orbit. Due to their relative sizes, any noticeable rock or asteroid would encounter issues of a 3-body-problem nature, and either hit Earth, the Moon, or be ejected.