My apologies. I used the term “Color Picker” when it should have been “Sample Color” (Shift X), which has a similar, but not equivalent function as the Color Eyedropper. Sample Color works within the context of the 3D Viewport in Texture Paint mode specifically (paint.sample_color), so I would expect the Eyedropper hotkey/quick favorite to work in the same Context. That is, if I’m in Texture Paint mode and my mouse is in the 3D viewport, I would expect the hotkey/quick favorite item to assign the current color to the same place as Sample Color since the hotkey is the color-specific eyedropper (ui.eyedropper_color), and the quick favorite assignment is very specifically the color picker eyedropper too, so the object eyedropper in the sidebar is not relevant here.
