True, although the mad monk had a snake, Quicklime, as a familiar, and the Count naturally had a bat. And Snuff, as drawn on the cover of the hardback novel, was an English sheep dog. Come to think of it, if you were out moving though a modern city, the only animals that wouldn't draw unwanted attention would be Snuff and Graymalk. If you walk around with an owl or raven on your shoulder, or you've got a bat or snake following you, those animals are going to draw a lot of attention. I suppose it would also depend on what those animals could do--what can a dog do that a cat can't, what can a snake or owl do that dog can't? What specific skills do I need to accomplish my tasks?, etc.
So, yeah, why not a dragon? The reason I chose a dragon is that if I wanted to close things out, a dragon would have enough power to do it. The dragon could also be smaller or invisible to most people. It would depend on how you wrote it.