I don't think that's possible in app designer. Here's are two alternatives that will require a little work but they're doable.
Idea 1: clickable axis
You could add a new axis in place of the drop down list (same location, same shape) and it can display the blocks of colors as in your image. That's easy using rectangle() or patch() or fill().
Then you can use ginput() to capture the location of the mouse click within the axes. Once you have that coordinate, there are several ways you could determine which rectangle the click was in. Then grab the color values from that rectangle.
Idea 2: colored buttons
I think I like this idea better. You could add a stack of small buttons (button group) that do not contain any text but are merley colored (as in your image). When the user selects a certain button, a callback function can assign the color value.
Idea 3: use a knob
In app designer you can use a discrete knob so the user can select a color. Next to the knob you could have a tiny axis or botton that changes color with the knob selection so the user can see the result.
If you get stuck while implementing any of these ideas, I'd be happy to continue the discussion.
