No, the handle and bristles of a tooth brush should be made up of different materials. The handle of a toothbrush should be hard and strong, while the bristles should be soft and flexible.
The bristles of a toothbrush are made up of nylon. Because nylon is a synthetic fiber which is soft for the teeth while strong and flexible for usage as it doesn't degrade in water or with the types of ingredients that are found in toothpaste.
The handle of a toothbrush is made up of plastic so that it can be injected into molds because it exhibits plasticity. The most common plastics used are polypropylene and polyethylene.