Sonia Chang-Díaz knows what it’s like to be ‘other.’ It has shaped her and her bid for Massachusetts governor.

Sonia Chang-Díaz is familiar with being “other.” In a Newton elementary school cafeteria, she was the girl teased for her lunch of rice and beans. She was the 10-year-old followed by a clerk in a Montana store while her blond-haired, blue-eyed cousin shopped freely.

Before legally changing her name in her 20s, she was the young woman — Sonia Chang — whose surname confused strangers who would awkwardly wonder if it was, in fact, a married name. “You don’t present as clearly Asian,” they would suggest.

“If you’ve never been on the outside,...