Is there a way to drag a public color into another object's color field?

what I am planning to implement are public color channels that other objects in the scene can use. When a channel color is changed, any object using that color will also change. Essentially I want objects to have shared colors

I thought a very simple way of doing this would be to just drag the channel color in the inspector to the object renderer’s color field, but it doesn’t look like it is possible to drag colors.

So is something like this possible or am I going to have to try a more complicated method?

Yes but not exactly you are saying. but you can make it with trick.
choose color for the publlic field. and in color picker you can save the picked color in bottom right side. and then choose it in another objects public field.