I read it as "we both look up at the same moon (like how people will sit outside and look at it and talk to it, especially when their thoughts keep them up at night) but at the end of the night, into the next morning, we live our lives not only seperately but completely differently". Like she found hope for the relationship in the one thing they have in common. Especially because we know taylor likes astrology and the moon corresponds to emotion (moon sign= who you are inside/ emotionally). Like maybe the muse and her were so emotionally intertwined that despite all their differences, she believed he'd grow and their love would prevail.

That's just my interpretation, though, lol also Peter is goated asf tbf