Welcome to the "why won't asset browser do this?" club. Population... a lot of us. So, I have learned that many types of things do not actually save in the asset browser. No parented meshes - only single meshes, materials and HDRI's (so far, that is all I have verified).
In the case where you want to save an image, I recommend saving it via saving a Material (which will, of necessity, include a pointer to the image in the blend file). It's a workaround, but it will work. The material will be saved in the library and a pointer to that file will be inside the material (because it'll be plugged into a node).