I think he's off a little in this part:
> Zoozve orbits one thing: the sun. It spends all day every day doing that. BUT Venus also has a teeny gravitational toehold on it such that it ALSO ORBITS VENUS AT THE SAME TIME.
> It’s a whole new category of thing. Something that orbits a star and a planet at once. Something that is not a moon, but also not not a moon.
> They call it … a quasi-moon.
> Astronomers had been speculating that such an object could exist for 100+ years, but this was the first time anyone saw one … not only in our solar system but in the entire universe!!
He then goes one to describe how weird the paths of quasi-moons can be.
Where I think he's off is that it was not that Zoozve was orbiting both the Sun and Venus that was new. Astronomers have for hundreds of years known of a case of something orbiting both the Sun and a planet (see below).
What was new was the weirdness of the orbit.
The long known example of something orbiting both the Sun and a planet is the Moon. The Wikipedia article on the orbit of the moon [1] has a plot showing a section of the Earth's and Moon's trajectories around the Sun and you can see from it that they both trace a convex shape. That's because the Sun's pull on the Moon is about twice as strong as the Earth's.
From the Sun's point of view both Earth and Moon are in separate orbits that have been distorted so that they are rounded dodecahedrons. They are like two cyclists racing on a circular track, taking turns passing each other on the inside but at all times both are turning left.
Compare to other (non-quasi) moons. Most (all?) of them are more attracted gravitationally to their planet than to the Sun. The plots would not be convex. From the Sun's point of view plots of their paths would look like something from a spirograph.
If alien astronomers knew of the Earth and Moon but did not know their sizes, so all they had was their masses and orbits, they would probably actually classify them as a double planet rather than a planet and a moon.
What disqualifies the Moon from being a planet under the current Earth astronomy definition is that the center of mass of the Earth/Moon system is below the surface of the Earth.
[1] https://en.wikipedia.org/wiki/Orbit_of_the_Moon