Their mother was chosen to become the Aspect of Justice when she was pregnant. As such, her children were both chosen to embody the same Aspect when the time came for it to choose a new mortal vessel. Touching the sword caused the Aspect's power to awaken within them, much like how Atreus (Pantheon) became the Aspect of War when he accepted the spear. We're not sure what happened to the mom (probably died off-screen) but Kayle and Morgana are both just Aspects now, albeit one embraces the role and the other rejects it.
The wings are weird though. I guess because they were already semi-celestial/divine when they were born, it caused them to undergo like a lesser version of Shuriman Ascension? Still trying to figure that one out, admittedly.