No, not every fur has a fursuit, the costume they wear to become a furry character. These costumes come in a variety of shapes, sizes, colors, and styles. There are realistic suits that look like the real animal is standing in front of you as if you were in a science-fiction movie and there are "toony" suits that seem like they just jumped out of a TV show. There are green husky dogs and blue wolves, there are brown dragons and red dragons, there's a grey cat and a tawny weasel. 

These costumes are of characters the people make. You can call them original characters or fursonas (a furry persona). To bring them to life, a person can make their own or commission one from a skilled seamstress. The process of making a fursuit is laborious and takes careful work and measurement. They can take from a month to a year to be made, depending on complexity. 

The customer sends a drawing of their furry character. This depicts the animal, it's species (real or made up), and the markings on it's body. Then the customer usually sends a cast of their entire body. This cast is made of duct tape and is used for precise measurements on their body so the fursuit is tailored specifically for them. The maker looks over the furry character reference sheet and begins to work on the costume.

Now, there are different kinds of fursuits as I've said before. Another kind of fursuit is digitigrade which means the legs bend in two places like a dog's hind legs to give the costume a more animal-like look. The other kind is a plantigrade suit which has straight legs like a human's. These don't give it the animal look, but allow easier mobility for the wearer. There are different sorts of fursuit heads like movable jaws which move when you speak or protogen heads which show the character's emotion digitally on a rounded screen where the face is. 

The wide variety of fursuits and fursonas allows everyone to be clearly distinct from each other. Instead of having to draw yourself or point out a person, you can draw a brown dog (TazD lol) or say "hey, who's that blue soda YouTuber" (Pocari Roo). With all this individuality, it's usually easy to identify furs by merely a description of their character. red fox with green and purple stripes who likes berries? That's Majira Strawberry. A black sabertooth tiger with white dots and purple hair? That's Gale. A lynx with purple hair? Strobes. A blue wolf with purple and green hair? Odin Wolf. That one dancing brown dragon? Telephone.