In my opinion, it is a combination of a time loop and alternate timelines. It clearly is not a 100% time loop otherwise there would be no hope, this event would continue forever and never be solved because there was never a Yuuta that was capable of solving it because there never will be a Yuuta capable of solving it.

Or in more terms relative to this show, the cat outright says he won't allow Yuuta to change history if he doesn't want it to. Yuuta has already changed this history. The interaction with Mikatan and Ito occurred two different ways in episode one. In this episode, the first time almost happened, but it was changed by Physical Yuuta's interruption.

It is a loop in the sense that the overall events will repeat, but what every Yuuta does can change the results of those actions. Like a hot wheels loop-de-loop racing track, it began somewhere, and however many loops needed are going to happen. But the race track will come to an end, it will end when he finally solves this problem. We are given no information on the beginning. The original poster I replied to was asking that question. I compared it to Harry Potter as an example but it really is different like /u/maleficarium mentioned. Given that this show does allow the timeline to change, it is entirely possible for there never to have been a first body takeover. The first Yuuta was just forced to live to the final day until he used what knowledge he could from that ghost life and went back in time to become the first spirit to body Yuuta. Remember time travel has nothing to do with going back to a body or not, our ep1 Yuuta destroyed the world then went back to right before he saw Ito's panties to repeat just that portion of the scenario. The end goal of all ghost Yuutas is to learn enough so that they can travel back far enough to reclaim another timeline's body, this would've been the first Yuuta's goal by the end as well.

But yes I do think the cat is suspicious.