It does not look like it is a vector illustration. It looks like you are trying to recolor a regular image or a raster image. This will not work.
You have to recreate the artwork as a vector image. Then you can recolor it. Now if you want to simply recolor the image without creating a new vector graphic. Then use Photoshop, there are several ways there to do it.
What is your end goal? Can we see the art? We might have more ideas.