Why are cloth-based characters so big in animation and games right now?

First it was Sackboy in the popular Little Big Planet game, then Coraline was full of living dolls and clothy creatures, now the upcoming film 9 seems to be populated by creatures made out of cloth.

Why’s the “fabric aesthetic” so hot right now in things that are animated, like games and movies?

Sometimes these things are based on the state of the art. Fur is expensive and doesn’t always work out (think Scoobie Doo movies). So when they find something that works they milk it.
At the time Toy Story came out, Woodie was cloth, but everything else was shiny plastic because that was the current problem solved.

Once something becomes popular everybody has to jump on the bandwagon. Truthfully, it’s just one more fad. If you ever figure out why some things become fads let me know.

I think it’s coincidence, mainly.
Coraline was based on the 2002 book, which features the button eyes.
9 is based on a 2005 short film.
Little Big Planet sprung from rag doll physics and they ran with it.

Dolls have been coming to life since at least Pinocchio.