I have an idea that involves a color changing mug that has one image before you put coffee in, and another after you put the coffee in. However all of the custom mugs I’ve seen to order have only black disappearing ink. Are colors other than black made? Is it possible to create multiple (changing) arbitrary images?
Great idea, but you aren't the first to think of it (but maybe you can do it better). :-) Multi-color and picture-switching mugs and other objects have been around for a while. But every color you add affects the cost and complexity, which is probably why you've seen mostly black. To do what you describe, you'd put on the first picture, then cover that with a picture using paint that switches between opaque and clear.
A wide range of "thermochromic" colors and changes are available. Some switch between color and clear, some change from one color to another, some display a range of colors. There are also various temperature thresholds, depending on what temperature you want the change to occur.
There are several kinds of materials that can be used to make color changing paint. Color Changing Mugs: How Do Heat Sensitive Mugs Work? explains the most common mechanisms. If you want to try it yourself, you can buy the paint (here's one link to get you started), or make your own by mixing thermochromic pigments with a clear binder. For example, here's an Amazon link to one pigment supplier. (I've never used either of these particular products).