I don't like this system for two reasons.
The first being that bookshelves should have a restraint system that attaches to the walls. You could probably improvise something, but the video lacks that element.
The second is that the alternating brick pattern is weak for an open faced box. That puts a significant portion of the weight of higher courses on the middle of the span of lower courses. You can see some of the lower levels bowing signicantly. Since the back is rigid, but the front can flex, that will increase the tendency to tilt into the room and makes the tipping hazard worse. Add in an old floor that is concave and you have a significant hazard.
I like the concept, but this needs some changes before it is safe.
Edit: I'll suggest potential improvements rather than just naysaying. You could make two different width boxes. A full width and something like .8 width. You would stack the boxes alternating full width with partial width. The full width box would need 4 alignment pins and 4 slots. The boxes would stack in line vertically, but due to the alternating widths would still lock adjacent columns together. The important thing is that the vertical walls would be close together rather than landing in the middle of the spans.
Then I would add a cap board that can be bolted into the top boxes and would be used to attach a L bracket to a wall stud. Yes, this decreases portability, but not crushing children is more important than convenience.