To be fair, he recorded that entire first album of theirs by himself aka played every single instrument on it. The rest of their work has been the band but he certainly could running "The Dave Grohl Band" if he wanted to. I think he said "My" the same way any person in the band could have said "My".
Clearly Zach Snyder is incapable of understanding awesome as well as understanding the concept of the alphabet. In this case, the H comes AFTER the O. In your case, however, the current usage likely still applies.