New "gravity force" analogous to magnetic force? I was watching Eugene Khutoryansky's physics vide

It makes sense that you had never heard of it in high school: this is actually a prediction of General Relativity, a subject complicated enough that many physicists will only study it in depth after they had already concluded undergraduate school. In this answer, I'll try to give you a general idea without going too deep into the mathematics.
First of all, General Relativity does not treat gravity as a force. Instead, gravity is understood as spacetime curvature: the reason things fall is because spacetime is curved, and hence things move in curved paths as a consequence. Think for example, about an ant moving on a rolled up sheet of paper: if the ant moves always straight ahead, it might still find itself back where it started, for example. It moves in curved paths not because of some force, but simply because it's moving on a curved surface. Gravity works in a similar way.
Now, the mathematics of General Relativity is quite complicated, and sometimes we want to avoid dealing with it in full detail. If the gravitational fields are not too intense (in other words, if we're interested in a problem where spacetime is not too curved), we can approximate the gravitational field with a force. For very weak fields, that is the Newtonian theory you learn at school. Newton's gravity is a weak field limit of GR. However, we can do slightly better.
Given GR, we can consider a simple distribution of matter, such as a possibly moving mass. Doing the calculations in GR and making a few approximations, we can get to something called Gravitoelectromagnetism. This approximation treats the gravitational field in a way very similar to how we treat electromagnetic phenomena. It is not as precise as full GR, but it is much simpler, and hence we often prefer to use it instead of solving the full complicated equations. In Physics, at the end of the day we're interested in comparing out computations to experiment, so as long as your experiment is not so sensitive that it will notice the problems of your approximation, this is fine.
In this approximation, you can see exactly the effect you mentioned: there appears a gravitational analogue of the magnetic field, leading to results just as the one you mentioned. Notice that this is not "a new force": instead, it would be more fair to say it is a consequence of the fact that gravity is not even a force to begin with. This effect is not a mathematical trick, it is real, as pointed out in the comments. The thing is it is really subtle, so it is quite difficult to notice it in most situations.

Adam is in a spaceship holding two magnets so that force exerted by his hands cancels their electromagnetic repulsion. Adam would see no electric force since magnets are at rest according to his frame.
Now Sarah is watching Adam from a ground reference and sees the magnets moving and hence they must exert an attractive electric force.
A force, analogous to a magnetic force produced by moving charge, is produced by moving magnet, moving muscle, moving planet, moving anything. Which is a good thing as forces that are balanced in one frame are balanced in other frames too.
To clarify a bit, the "magnetic force" for moving magnets is electric force. Which has been known since 1905, at least.