It's not a background, it's just another SVG object. The rest probably doesn't make a lot of sense. It's just not how things work and I can't figure what would be the problem with separate objects and making them interact in ID. It supports blending modes and simple masking, so what would really be the problem?
And the black element is not an SVG, that's why it will not get exported even if I select it together with the other 2 element, as an SVG. Only gets exported as .ai or .pdf. Before exporting it's a frame with a fill, not an SVG.