Every single person has their own version of a shinigami.
In every anime I have watched(that have a shinigami) have all different shinigami.
I'll tell you what I mean:
In Bleach, shinigami play a big part in the plot, but that's not what I'm explaining. The shinigami in Bleach wear black robes, use kido(i dunno what it would translate into English as, but something along the lines of 'demon arts') and wield katana called Zanpakuto. They are in charge of sending souls in the world of the living called Pluses(only mentioned once, the other times they just call them souls) to the Soul Society and purifying Hollows, souls that have lost themselves(basically, I think).
In Inuyasha, the shinigami(called shikigami in Inuyasha) are not exactly shinigami. They are those paper puppet things(whatever you call them), but I'm just going to include them into the shinigami category. So, those shinigami are short girls with powers like a priestess'. There are only two, and they act as Kikyo's bodyguards(or something like that).
In Deathnote, the shinigami are monster-like creatures that carry a Deathnote so they can kill people when they write the person's name down. Each shinigami is as unique and different to each other than humans. They usually stay in the world of the shinigami, but they occasionally go the Human world when they need to.
In Full Moon o Sagashite, ok, I haven't actually watched or read it, but I did check out the shinigami when I heard that there were shinigami in it. So anyway, the shinigami in Full Moon are silly and can be flustered at times(from what I have seen, which is not a lot) and the two shinigami you first see sort of mess up a lot and call themselves the Ramen Noodles or something like that. The shinigami are definitely more childish in this anime. But, then again, I haven't watched or read a lot of it, so I may be inaccurate on this one.
So, now I'm hoping that you know what I mean by everyone having their own version of a shinigami.
Feel free to correct me if you find any information that is inaccurate, just leave a comment.
Wow, it's a really long explanation. I wouldn't be surprised if you didn't read the whole thing.