Your native language - that's the language(s) you grew up with, of course, you might say. That's also what most translation agencies and sites mean when they ask about a translator's native language(s). Frequently, more than once choice is permitted in the "native language" field, acknowledging bi-lingual childhoods. But what about bi-lingual adulthoods?
The German word for native language is "Muttersprache" -- mother's tongue. I have met German translators who insist that one's native language is, indeed, one's mother's language, even when one grows up in a household where the father speaks a different language. That seems to me only true if the mother provides most -- if not all -- childcare during a child's first 2 or 3 years of life. While that is still relatively common in many German-speaking countries (where maternity leave is generally longer), in the U.S. it's often the nanny or daycare provider who teaches a child his or her first words. Some parents even deliberately hire a caregiver speaking a different language, so that their children can grow up bilingual, even if both parents are monolingual.
A friend of mine was born in Argentina, spent his toddler/preschool years in Israel and started school in Austria, where he remained until adulthood. Through all these moves, the language at home was German. He did attend preschool in Israel and hence learned Hebrew as his second native language, but didn't use that language again after age 5. When he returned to Israel as an adult, he found that some of his forgotten Hebrew came back, but with the vocabulary of a 4-year-old.
I, on the other hand, grew up in an entirely monolingual family (albeit with the more ex- and intensive foreign language study required in Austrian public schools). Many agencies and translator sites would therefore consider German to be my native language. However, my entire professional education and most of my adult life were conducted in English, and that continues to be my language of daily discourse. I therefore find it much easier to translate into English, and the results tend to be more idiomatic, as well.
This is particularly true in my specialization, IT. When I left Austria more than 25 years ago, I had never actually used a personal computer. Here in the U.S., I learned not only to use computers, but also to repair, maintain and document them. While I can understand computer texts written in German, I am much more fluent in English "computerese", and therefore translate from German to English, rather than the other way around.
There are any number of circumstances in which someone might initially grow up with a particular language, but be immersed in an entirely different language community as an adult, eventually acquiring a better facility with that new language than he/she had with their initial one. This happens to native peoples and migrants around the globe, if they move to a place where their childhood language is not spoken.
It seems to me a much better term would be "dominant language", meaning the language one is most fluent in. "Native language" suggests something one is born with -- a fact that doesn't change later in life. "Dominant language" acknowledges that for some people the language in which they are most fluent may change during their lifetime.
Translators, in particular, frequently end up in that profession precisely because their linguistic environment changed over the course of their lives. Some people continue to conduct their daily lives in their language of origin, and feel more comfortable in that language. But some switch languages more thoroughly, even writing literature in that new language. (Full disclosure: I write children's and young adult stories in English.) If one can be a novelist in a language one didn't initially grow up with, one can surely translate into such a language, as well.