If you look at the 1st volume of M. Saito Sensei's "Traditional Aikido Series" you will find photos of "Partner practice with the jo" exercises. These are 8 in all and are applications of the 31-no-jo kata.. They are very short and basic. If you look at the Original Japanese text, it clearly spells out "Jo-no-awase-ho" (see attachment 1).
In the videos of M. Saito Sensei about Aikijo, the applications (bunkai, if you will) of the 13-no-jo and 31-no-jo are also referred to as "awase".
I was fortunate enough to attend courses conducted by Yasuo Kobayashi Sensei and his son, Hiroaki Kobayashi Sensei. When they taught Aiki jo, they first taught the 8 awase shown in Volume 1 of Traditional aikido. FIrst, it was performed as synchronized practice, with each one doing the same movement (the defender). This is similar to the manner of performing renzoku-dosa (see below). Then it was performed with a clear "attacker" (uchijo) and "defender" (ukejo), as depicted in Saito Sensei's book (see attachment 2).
Y. Kobayashi Sensei also taught the renzoku-dosa as shown in the video you posted. Here, both partners do the same movements. If you look closely, they are based on the 20 Jo suburi, and there is a switch from doing it on the left side to doing it on the right side. This helps in developing the ability to perform jo techniques on either side.
Y. Kobayashi Sensei also taught awase of the 13-no-jo but not in the manner shown in M. Saito Sensei's video. Instead, both partners did the same kata but Ukejo (defender) was one "step" or "beat" behind Uchijo (attacker). You can see this in the video of Hiroaki Kobayashi Sensei demonstrating the 13-no-jo:
Hope this helps..