I think it's important to remember that you don't need magic to be able to have multiple identities. Forming multiple personas is a common response to trauma or other emotional stimuli, and Shallan's plurality is, by the scale of traumatic plurality, fairly mild. Her experience fairly closely matches my own (non-traumatic).
While Lightweaving is probably playing a part in her development of the personas, it's clear to me, especially in the later chapters of OB, that fundamentally these three aspects of her are just a part of her. They aren't dependent on Stormlight or the Nahel bond or anything else other than Shallan's own conceptions of herself.