In fact, their packaging is pretty minimal. Usually just a clear plastic tub with the product name, ingredients, and use information. Some of their products have small “not tested on animals” and “recyclable” logos. That’s it. Their social media profiles do mention “eco friendly products” in their bio/about pages, but they’re not really pushing this message in their posts.
One news article I found says that Ecoco describes itself as using “globally sourced ingredients [that] are nearly all naturally derived, delivering great personal style without the carbon footprint.” This is the only time I’ve seen this carbon footprint claim, and I can’t find any other details about what this means, so again I don’t think there’s anything to this.