Not any more than it’s fair to assume that, by showing the Barge of the Dead, or the Miranda-class, or the Oberth-class, or the Galaxy-class, or the proto-Klingons, that VOY: “Barge of the Dead”, ST II, ST III, TNG or TNG: “Genesis” took place in a reality with different class ships or people.
The Mirror Universe question is a separate one, to which there really is no good answer because we’ve only seen crossovers from the Prime Universe to its Mirror Universe counterpart. A bigger question is whether or not the Mirror Universe we saw in PRO: “Broken Mirror” is the same Mirror Universe we see in DS9 because there the Terran Empire seems to exist again.
But that still doesn’t mean DIS is in an alternate reality.