Merriam-Webster dictionary defines felt as "a cloth made of wool and fur often mixed with natural or synthetic fibers through the action of heat, moisture, chemicals, and pressure". The fibers are pressed together by various means rather than sewn or woven together.
There is a wide variety of felts and and equally wide variety in cost. One of the most common "felts" are craft felt, easily found and commonly used in craft projects. It is usually made of synthetic fibers and is low cost. The fibers come apart (resulting in pilling or fraying) with light use.
Art felt is made buy hand of wool, silk, cotton or synthetic fibers. It is an artisanal product that is suitable for clothing, hats and three dimensional objects. It is more dense than craft felt but not dense enough to wear well in a bag.
Industrial felt is a dense, durable fabric used in industrial applications such as shoe insoles, a roofing layer, machine washers, items for sound and/or vibration dampening. It is made of lower quality wool and synthetic materials. It is durable but lacks the finish of wool designer felt.
Wool designer felt is made of wool from merino and karakul sheep. The fibers are forced into a dense felt using a steam and pressure process. The density and consistency are perfect for bags. It takes dye well and is available in a wide palette of colors. I could buy less expensive felt, but it would not wear as well nor finish as well as wool designer felt. Quality matters.