The quest does not stop of pop-up. It does not give anything after completing the first time. I took some pictures to demonstrate the glitch. Thanks in advance!
Intentional probably. What if you got a new pet and Kenta was like "sorry I already made your husky speedy I'm not doing it again".
Yea it's intentional. Any new pets you get won't get the speed boost until you bring them out and re-complete the kenta's advice quest (what you've been doing in the video, not doing the whole quest all over again). It's needed if you complete it with only 1 pet, then decide to get the 3 pet skill later and give all those pets the speed boost.