The kata in this video shows the basic applications of
kaeshi waza, meaning counter techniques. The history of
kaeshi waza goes before the foundation of aikido, it is already a known concept in older martial arts like
judo, etc. Here is a link to Judo terminology.
It is of course no surprise as Aikido has its root in older budo arts. Shihan Stefan Stenudd (7.Dan) has a nice list of kaeshi applications for Aikido in his website, showing which technique can be countered with which: Table of Kaeshiwaza techniques
I also add a screenshot of the table, so we will not lose the information if the website goes offline.
As the last point why some Aikido schools don't teach Kaeshi waza and some do, is difficult to answer. I have once heard from Shihan Adrien Halm (8.Dan), who is a disciple of Hirokazu Kobayashi, that O'Sensei teached
Kaeshi waza before the World War 2. After the war it was removed from the art because of the martial arts ban between 1945-1950 in Japan. However I could not find any source or documentation to support this argument. It might be a speculation.