Thanks @Alexey Popkov. That is part of the story, I would like to make it such that it can be interpreted again, like the things colordata makes:

i.e. you can copy the output and add [value] to it, and it will execute... This seems to be a bit trickier to make...

I am trying to dig through this myself, but I do not quite understand this. It might be related to the Low-Level Notebook Structure described also a bit in the tutorial Representing Textual Forms by Boxes. If you look with CMD+SHIFT+E on those structures you can see things like InterpretationBox and TemplateBox.

ColorData["Rainbow"]

InterpretationBox makes a formatting structure whose interpretation is not based on its appearance. For example, lets compute these:

Plot[Sin[x] Cos[5 x], {x, 0, 15/4}, PlotTheme -> "Minimal", Filling -> 0]
Integrate[Sin[x] Cos[5 x], {x, 0, 15/4}]

and now pack them as

Now the last output is interpreted as the expression in its second argument:

I doubt this is really helpful. I also would be curious about user-built case for such structures.

Perhaps you also need to set Format[YourOwnFunction] := ....... ?