I feel the same way. For me it makes totally sense that Erie kept the CD in order to learn the Japanese song because he wants to impress Mutou when they meet again in real life, somehow. Furthermore, Mutou kept Erie's 'wedding' ring although Erie expressed clearly that this ring means so much to him and he wants to have it back.
Eerie has been emphasizing throughout that Mutou is the embodiment of pride though. Mutou is too proud to "plea" (and unlike Sword and Sen, Mutou doesn't have anyone he can rat out to trade for his own life, at least, not without losing his "dignity"), so that's why Eerie knew Mutou was lying and that Mutou really did get executed. I think Mutou kept the ring partially to convince and comfort Eerie and partially as a promise to meet in the afterlife or next life, because as Eerie said twice in their last moments together that it's just about "who goes first." It seemed like Eerie understood all of this. Just like how Eerie treasured Norman's ring, he now treasures Mutou's CD. Unlike with Norman, whom he knew fell for Van and he would have to force himself to forget about him, Eerie seems to think of Mutou as his last love (since to him, Mutou just "went first."), so he's perfecting his Japanese singing in preparation for meeting Mutou in the afterlife. Although their time together was short, Eerie's really perceptive so he understands others well (like how he knew Norman fell for Van and tried to help Norman find the will to live on).
The story continues in Under Grand Hotel: Cross Over - http://www.mangago.zone/read-manga/under_grand_hotel_cross_over/

is this the end???? so did moutou died or no??? its impossible that he is not, right? maybe he meant by (until they meet again, he means in heaven or hell!! so please someone tell me is this the last chapter or no??