First Language vs Native Language
For example, my sister is born in Vietnam, her first language is Vietnamese. She soon moved to America by age 5, being a Dec. child. Yet her native language is English. This is correct because she expresses and think in English. Her mind speaks in English. She identify herself with the English language.
First language is a mother's tongue. A kid before five could speak in any language, and it will be their first.
But the language that the child later learns to express themselves is their native language. Native deals with your instincts. What comes natural.
Now she is learning Vietnamese. It not her native language but a language in her mother's tongue.