Trickier to answer then you'd think, because presumably the phrasing excludes all roughly human-shaped aliens and robots. I wonder about John Henry from The Sarah Connor Chronicles. It would probably be cheating to include him, for despite him being actually in the form of several boxes, it's the human-shaped extension and the actor of same who sells us on John Henry as someone to care for.) Similarly, someone like Chewbacca from Star Wars or Novice Hame or Brannigan from Doctor Who would be out because despite being covered by fur from head to foot, they come in human size and shape. So, let's see, beloved characters who really don't appear in human form.

1) Pilot from Farscape. Aka the one everyone is thinking of when they tell Farscape sceptics "the muppets will make you cry". Gentle, occasionally snarky, with as much tragedy in his life as any of the human-shaped characters. Pilot is immensely huggable (complicated by the fact he has a lot of limbs), and that's a fact.

2) Rygel from Farscape. As if I'd leave my beloved Dominar out. Rygel is Pilot's opposite, starting with the fact he's probably the oldest character on the show until Noranti in s4 (and Pilot is very much a teenager for his species, though we don't have confirmation of that fact until s2). He's greedy, eternally hungry, very shrewd, great at negotiations, actually very dangerous if you underestimate him due to size, and over the course of the show finds himself starting to care for certain people despite himself.

3) Matthew from Sandman. A borderline case as he used to be human before dying and becoming a raven in the Dreaming, but that's his Swamp Thing backstory before Sandman starts, and we never see Matthew as anything but a raven in Sandman (and wouldn't know he once was human if not for two or three lines of dialogue in ten volumes), plus the saga makes the point that his form is also his nature now when he finds himself feasting raven-style. Hence I say he counts. Matthew is one of my all time favourite sidekicks as well, no-nonsense, loyal but asking questions anyway if he doesn't understand something, and putting up with a lot from having to yell driving instructions at Delirium to being sent out in tandem with a serial killer with teeth for eyes.

4) The cat from Coraline. It's very evident from his works - both comics and novels - that Neil Gaiman likes cats (and cat godesses), and my favourite of his cats is the one from Coraline. The cat - who refuses to be called by a human name ("I know who I am") - is the one ally Coraline has (in the novel; the fact this is changed for the film makes for a somewhat different story), and so very, very feline both when silent and when speaking. No one's sidekick, I'd call it Coraline's occasional companion - when it chooses to be.

5) Fuchur the luck dragon from The Never-Ending Story (again, book, not film; the design for Fuchur was one of the many things Michael Ende disliked about the movie, because the description for Fuchur in the novel is pretty definite - think Chinese style dragon). Fuchur was the first friendly dragon I encountered in fiction as a child, and no less impressive or awesome for this; when Atreyu finds Fuchur escaped Ygrammul's net along with himself I was so relieved like you wouldn't believe. Fuchur is still my favourite dragon, come to think of it.