What makes a bisque a bisque and not a soup?
You get 10 points for bringing up an interesting question, but lose nine of those points for asking the wrong question. You should have asked, "What distinguishes a bisque from a chowder?" A bisque like a chowder is a soup, as you well know. But what separates one from the other primarily is lumps.
In traditional French cooking, a bisque is a thick, smooth shellfish soup in which the seafood has been puréed. The most well-known are lobster bisque, crayfish bisque, crab bisque, and shrimp bisque. A chowder (even though you didn't ask) is a chunky stew, and also most often made with seafood.
The preparation of a bisque is no small achievement. It involves marinating the shellfish of your choice in a sherry-infused court-bouillon overnight, cooking the fish in the court-bouillon with tomatoes, removing the fish and pounding it to a purée in a mortar and pestle, passing the purée through a sieve back into the bouillon and simmering it for a couple of hours. Finally, the hot soup is whisked into a tureen where three (not two, not four) beaten egg yolks are waiting, after which some cream and brandy or sherry is added.
We have not come across a recipe in reasonably modern cookbooks that take this approach, however. Most thicken the soup with rice and purée everything in a food processor or blender. Some very respectable cookbooks even leave the fish in chunks in the soup, in which case it is a what? That's right, a chowder and not really a bisque after all.
In earlier periods, say, the mid 1700s and before, one might stumble upon quail or chicken bisque, in which case the meat was not puréed, but crusty bread was added to the soup and cooked to death in order to provide the right consistency. The origin of the soup and the name, by the way, is thought to come from the Spanish province of Biscay, and the original bisques were spicy soups with meat or game garnished with crayfish.
Given centuries of evolution, is it any wonder these days that the word bisque may be associated with any thick, creamy fish-, tomato-, or vegetable-based soup, right down to the Campbell's Tomato Bisque (which does have lumps) available in many supermarkets?