This is one of the places where the language has changed over time.
Fulling used to apply to only woollen (aka, wool spun a specific way) cloth. It now refers to the change that happens in wet finishing cloth.
Wet finishing is the process of washing and agitating the cloth. This works on every type of fibre, plant or animal. But it works differently on different fibres.
Most animal fibres (wool, alpaca, bunny, etc) will felt a bit when fulled, although the degree of felting depends on the specific animal and the way the yarn was spun. It closes up the gaps to make a lovely cloth.
With cotton, wet finishing acts simularly. The
water and agitation relaxes the cotton and lets the fibres poof out a little. So the finished result is a lot like fulling. The cloth shrinks, the holes are no longer able to let a finger pass through.
(almost) All fabric benefits from wet finishing, but wool and animal fibres show the most dramatic change.