I only heard about double chin which means a fold of fat under a person’s chin, that looks like another chin.

I had never heard “treble chin” before. It’s not a frequently used expression. Usually we stop counting once we hit two chins!! The speaker is really trying to emphasize the fatness here!

it means having multiple chins folded over one another (due to being fat).

The usual expression is double chin. This is how people sometimes describe that part of the body just under the chin when a person is overweight to the extent of having a fold between the tip of the chin and the beginning of the neck. If someone were very overweight this could be described as a 'treble chin' - where the fold is in three parts.

Double chin (extra fat in chin area) Triple chin Treble chin? No idea. Typing mistake?