A native speaker would probably mean a person who has learned English as a language at an early age and probably (?) at the time, he/she was learning English as the sole language in a home, where English was the sole language.
For example, my grandmother grew-up in a home where German and English were both spoken. She learned German and English as a child, but as an adult, she never spoke German. She spoken only English.
She was a proud German-American and a very good woman.