Due to changes in instrument technology, the terms for categories of instruments such as woodwind or brass no longer refer to the material. Originally woodwind instruments were all made of wood, but the more important defining factor was the method of sound production, which involves a vibrating reed. (Yes, even on a flute the lip/tonehole edge are actually acting as a reed). It was discovered that wood was not the defining factor in the sound, and at some stage both flutes and clarinets started to be made from brass with very little (if any) difference in sound. Metal clarinets are quite rare today, but were very useful for military bands in countries that had hot or humid climates that would cause wood to crack.
Brass instruments have a different type of tone production, which involves “buzzing” the lips into a cup shaped mouthpiece. Again, they do not need to e made from brass – the earliest forms of brass instruments were made from animal horns.
These days both woodwind and brass instruments are made from other types of metal and plastic, but they still keep their characteristic sound due to the tone production process.