A small asteroid could orbit Earth at the distance of the Moon in the L4 point, 60 degrees ahead of the Moon, or the L5 point, 60 degrees behind the Moon. Orbits in L4 or L5 points are called Trojan orbits.
The lunar L4 and L5 points are also considered to be stable points for artificial space habitats, so explaining the name of the L5 Society.
But I have the impression that in Trojan obits the main object, in this case the Earth, should have many times the mass of the secondary object, in this case the Moon, which in turn has to have many times the mass of the tertiary object in the L4 or l5 point.
As far as I know the only objects in the L4 and L5 points of the Moon are concentrations of interplanetary dust called the Kordylewski clouds, so faint that though they were first detected in 1956 they were not confirmed until 2018. Their mass must be minute compared to that of the Moon.
Perhaps the Earth could have two moons of equal mass in the same orbit, 60 degrees apart, one moon being in the L4 point 60 degrees ahead of the other, and the other moon being in the L5 position 6 degrees behind the other. But I don't know if such a situation would be stable, and I know of no examples of such a situation.
It has been claimed that two planets with similar mass could be stable in the same orbit if they were 60 degrees apart.
Two planets with similar masses can also share the same orbit if they orbit 60 degrees apart. This means that each is in the other’s L4/L5 Lagrange point. This kind of configuration comes out of our computer simulations, and we expect to find one of these setups among exoplanet systems.
If that is correct, two identical mass moons ought to also be able to share the same orbit, spaced 60 degrees apart.
Here is some additional information.
The PlanetPlanet blog has a set of posts called The Ultimate Solar System, designing solar systems which have as many habitable planets as possible.
The post The Ultimate Engineered Solar System designs a solar system which doesn't have single planets in each orbit, but rings of planets in each orbit.
It seems that a solar system could have a number of planets sharing the same orbit, as long as the planets have equal mass and are equally spaced in the orbit. The source is this paper:
Apparently such a ring of planets could be stable with seven to forty two planets in a single orbit.
And what is stable for a ring of planets around a star would also be stable for a ring of moons around a planet. Except that the gravity of the star would be a perturbing factor.
The Moon has a mass of 0.012300 of the Earth's mass, and the Moon's orbit with a semi-major axis of 384,399 kilometers would have a circumference of approximately 2,415,248.1 kilometers if it was circular.
So if there were 7 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.0861 Earth mass, and they would be spaced 51.4285 degrees, or about 345,035.44 kilometers, apart in their shared orbit.
So if there were 42 moons at the distance of the Moon and with the same mass as the Moon, they would have a total mass of 0.5166 Earth mass, and they would be spaced 8.5714 degrees, or about 57,505.904 kilometers, apart in their shared orbit.
In another post, Cohorts of Co-Orbital Planets, it was proposed that arcs of planets could share stable orbits and they didn't need to be complete rings.