Things to check for other than groups are clipping paths, compound paths and using the pathfinder palette and playing around to see what happens. Generally clipping paths have to be removed and compound paths have to be released to edit individual shapes. Even then, if the shapes were created using larger shapes (for example like the rabbit head and ears being one shape, and the body, and the arm being all separate shapes.. plus the mouth is a line with a fill, and I can't tell about the eyes and nose here) when released, the clipping paths and compound paths will change colors of the overall page/screen objects (which might freak you out, but you must dissect the collective.
Prior to doing this, it would be better for you to play around with these features (grouping, compound paths, clipping paths and all features on the pathfinder palette) with a few shapes that overlap and are filled with clear, easy to see color fills and line strokes. Apply something, and undo it. Repeat until you understand until you can pull apart your shape. It appears your object is vector, or it would not have the handles or blue lines.
OR - you could use the paint bucket and be sure to hover over one particular area, I'm not sure if you're doing that based on the post.
I don't think you can vectorize properly, its just either automatic or done manually. Of course, a manual product can anticipate needs better, thank god.