Yes, I'm surprised that I can still login to my old server. It is confusing.
Correction: I was wrong about the files. It appears that all my files on the old server were successfully copied over to the new one. (The only discrepancy between the two servers is that a new domain that I began hosting after the server switch shows up on the new server, but not the old one. This makes sense to me.)
But I'm still wary about deleting my old files from the old server. For all I know, those files are somehow, mysteriously, dynamically linked from the new server via some obscure process that I don't understand. For all I know, if I were to delete them, Badness would happen.
Perhaps this is a tech support question?