During the live show by ZHIEND, Otosaka Yuu remembers his time as a guinea pig in the research facility, where he helps his brother jump back in time.
The thing is, we later learn that his brother jumped back countless times (separating the guinea pig timeline from the current timeline even further).
How could Yuu have those memories?
Unfortunately, the answer to that is that there isn't an answer. In episode 11, Yuu himself asks this question to Shunsuke and Dr. Tsutsumiuchi.
This happens at 4:54 as seen below:
Some argument over this same topic took place on the MyAnimeList forums here.
In summary, that thread has a few different ideas on how it could have happened but no concrete evidence since the show didn't provide any. If you recall, throughout the series there were several hints of two of them remembering they had a brother. His memories finally returned when he was listening to the live performance of the song incidentally named "Trigger". This was supposedly the first time he had heard it, but he could remember listening to it in the other timeline.
Another theory from that thread is that Sala had an ability to restore people's mind with her voice, as she did with Nao's brother, and upon hearing her live performance his memories returned; however, this is unlikely because he previously heard her sing in the hospital room and was unaffected.
As the final episode has now been aired, it is clear that this will be left open as a plothole with viewers left to interpret it as they please.
The song Sara Shane wrote for Yuu, "Trigger" is the same song from the timeline where nearly everyone were being imprisoned and perished. This was not the only time that things were noticed to be off by Yuu. It is mentioned several times that there was a void that he either could not figure out. His experiences coupled with the time that passes and then the introduction of that song that aligned with the timeline of that memory, "Trigger"ed him to regain all of his memories.
As mentioned in Michael McQuade's answer, I think that she does have that power. It's just that she can give people their memories only if she sings for that specific person. What it means that she knew that Yuu was in the crowd and sang for him specifically and he was affected.
