Found it
I see what you mean. What you see there, though, it's not an addon that adds more moons to the default program. The number of moons for the gas giants is the same as in real life in that video, and as distributed with the program. But, since they show moons orbiting for a year, defined as the time it takes to the gas giant to orbit around the Sun once, and this in a few seconds of the video, they accelerate the time in SpaceEngine so much that the same moons appear multiple times per second, this video artifact makes the moons appear as they were many more.
You can do it yourself, by accelerating the time with the "K" key so much that moons seem to be multiplying.