I'm not sure I understand correctly what you want to do.
... adding 2 of the same image ... Meaning you made a virtual copy? Then yes, every virtual copy can have it's own color label.
Please explain in more detail.
I would like to be able to put separate colors on the same image if it is in 2 different collections. Say I have one image in my stock collection and I want to mark it with green. Can I mark it with red in my website collection? It seems like when I color code one image it changes it in every collection it is in. Hope this makes more sense. Thank you.
You're missing the idea of the collections. You are adding the very same image (a pointer to it) to the collection. Thinking of that, how can it have 2 colors.
If you need it to have different colors you then you need to make virtual copies of it as a workaround.