I don't think Jugemu was a reincarnation. I think that the problem stems from the fact that they look alike, but according to Jugemu, he was born somewhat before Edo period or during the Edo period and it seems as the other guy's story also happened during Edo period. So if Jugemu was already alive he couldn't have been a reincarnation.
Also, usually in mangas if a reincarnation were to happen, some sort of acknowledgment from other characters is seen. Like mentions of their similarities and things like that. And in this case, where it involves a Shinigami, even more.
The part about the name (for what I understood from the manga) was that because his name worked as an incantation of sorts and also because it was said in its entirety by a Shinigami no less, is like the incantation started working again and he began a new very long life. (Again, this is what I understood) ╮(￣▽￣)╭
Can someone please explaine to me these two parts I didn't understand.
1. How is he a reincarnation?
2. How does the shinigami saying his name related to him being alive again or like one of commenter said "he started his 400 year life again"?
╮(￣▽￣)╭