The Soul King is shown in the Thousand Year Blood War arc, though only briefly. Not much is explained about him, so things like what he can do and what his job is are a bit vague. Is he a Shinigami? If not, what is he?
Like Madara mentioned, it's really impossible to say at this point. From what I remember from that brief moment in the manga, the silhouette of the Soul King kind of looked like Aizen's pre-butterfly state.
Aizen also called "it" a "thing," so it might not be a Shinigami.
In Chapter 611, the Soul King was revealed to be
However, it was not stated anywhere in the manga whether or not he was a Shinigami or a Quincy. In Chapter 615, the purpose of the Soul King was explained. To quote from the wiki,
The Soul King's sole purpose is to regulate the flow of souls in and out of Soul Society. Without his existence, all known dimensions connected to Soul Society will begin crumbling into non-existence.