Monolingual? Bilingual? Oneandahalflingual?
Bilingual, but not fluent in the second language.
I’d say monolingual. Bilingual doesn’t care where the language is learnt. It just means “I can speak two languages.” This guy can’t.
For instance, I have four years of barely remembered high school Spanish. To call me bilingual is a generous stretching of the the word to put it mildly.
What do you mean by ‘growing up with’? I agree with Enderw24s point that it should not matter where language skills come from, but in terms of how it affects your identity I can see that it does matter whether you learned some passive skills in Spanish in HS or from your grandparents when you were very young. Either way, though, I probably would not refer to someone as bilingual unless spoke two languages really quite well to the point of being near-native. For a lot of people, the standard is lower though. Here in Canada, I find that the level of English spoken by Francophones and especially the French spoken by Anglophones that suffices for them to claim bilinguality is quite low.
I also wouldn’t classify someone as bilingual unless they were almost equally fluent in both languages. It doesn’t matter if they learned the other language as a child, and forgot much of it, or learned it imperfectly as an adult.
My native language is English. I speak Spanish reasonably fluently but am not close to bilingual.
Merriam Webster says:
Bolding mine.
On the other hand, they say of monolingual:
There really is no single word that means fluent in one language, but speaking another poorly.
Good question. I grew up in US with French (only speaking with Grandparents and to some extent with my father) – Yiddish was off the table, since my father’s side of the family was entirely non-practicing and non-identifying, and, anyway, my grandfather didn’t speak it at all, and nobody except my Grandmother could read Hebrew.
While I’m fluent and worked in France for roughly two years, and studied in college and grad school (no language courses, just literature courses taught in French), I’m not a near-native speaker or bilingual. Accent is very good, can understand the radio, pretty much anything that doesn’t come from a movie with 1940s gangster slang in it (although I guess I know that slang better than the very latest slang), and taught advanced French as a grad student for a few semesters (before discovering the boss was a total bitch and came back to teaching Comp), but I wouldn’t pass the state exam as near-native or better.
No point in that little tale, just an anecdote.
No, there isn’t, but we now know that growing up with a second language changes brain development, even if you don’t end up using the language later.
But that’s not what it refers to. It refers to people like me. I speak English very well. I learned grammar, spelling, and syntax in school. I can read and write English fast and accurately. Formal English is easy for me to understand, and I can understand technical writing. If I had to class it, I’d call myself a master, but not a grandmaster, so to speak.
I speak Hindi well, but not great. I never learned grammar or syntax or spelling. I never went to school for it, but I took after school classes. I can read and write Hindi but you get fast or you get accurate; I can’t do both. Formal Hindi is very difficult for me and technical writing in Hindi is way beyond my skill. I guess I’d call myself an expert.
That certainly doesn’t make me monolingual!
There is a real difference between people who grew up with a language but don’t really speak it and people who have studied a language as an adult. The former generally have close to perfect understanding of the language but can’t or won’t speak it at all, where as the latter have a much closer correspondence between their speaking and comprehension. They may also understand written language better than they understand spoken language. The two situations certainly need different words.
For the type of people described in the OP, I find “second-generation” works in most cases.
Slight hijack: What do you call someone who can’t speak either language fluently?
I run into these people all the time - Koreans who moved to the US when they were in their early teens. They speak Korean with an American accent, and English with a Korean accent. Accent aside, their command of either language is imperfect (for example, they often use the wrong article for a noun, or get their verb tenses mixed up). I always wondered if there was a term for this.
I would call someone bilingual if they speak two languages fluently. One of them may still be their primary language, in which their fluency might be slightly better.
Someone who has equal fluency in two languages, especially if they learned both as a child (they have two first languages, so to speak) I would tend to specify as “bicultural” or “culturally bilingual”. I, for one, am bilingual (in French and English), but not culturally, since I identify as francophone.
Sometimes I describe myself as speaking two and a half languages, on account of being in the process of learning Italian and remembering some Spanish from a long time ago. But it’s more of a joke, since I’m nowhere near fluent. And you’re right that there isn’t a single word to describe this situation.
I have a friend in this category. She speaks “Home” Korean fluently. Anything that has to do with running a household she knows, but on other topics she is lost. Her Grandmother speaks no English at all, and can only communicate in Korean.
We mulled this very question over for years, until one day I realized that her Grandmother had never finished school and had sucessfully run a household and raised several children; but didn’t know any more Korean than my friend did. Grandma spoke fluent Korean, with about a fourth-grade education, and so did her Granddaughter.
When my friend is planning a menu or doing chores around the house, she thinks in Korean. On any other topic she thinks in English. Bi-lingual IMO.
Canadian? 
Eh?
I would classify someone as “-lingual” if they can get by in that language. If you understand enough and speak enough to live in that world - can ask for help, buy items at the store, understand the PA announcements at the station so you don’t miss your train, hear teh news, rea the newspaper, etc. Diction or vocabulary would not need to be exmeplary.
SO what do you call someone who does not speak a language well?
I’m going to plunk for fluent vs. affluent… 
“He does not speak well, but he has a rich vocabulary.”
and if you need to use a small computer to help does that make you eFluent?
A linguistics professors of mine called this pernicious bilingualism, but that may have been his own term.