This may sound strange, but say you're talking to a friend on the phone, or an
equivalent. You want to ask if they have a new game or say, got the new
character in a game. I realise you can ask 'have you received', but is there
any way besides 持つ and ある? Like a verb to 'own' or to 'possess'?

If I'm not making sense, say if I asked you if you had a car. Would I just say
あなたは車がある？ What about a new character in a game? How would I say 'do you own
him' or 'have you got/received him yet'?
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word-choice english-to-japanese
How to say "do you have"
I suppose these could be used for gaming. Remember to use the right tense.

> 仲間に入れる - make a friend out of  
>  仲間に加える - to add into your group of friends  
>  手に入れる - often used when finding items in game  
>  見つかる / 見つける - to find/be found transitive/intransitive  
>  ゲットする - english loanword for "get" suru verb  
>  プレーする - english loanfor for "to play" hear it used in sports a lot. Maybe
> for videogames too  
>

There are a lot of loanwords in both videosgames and sports.

more general words for purchase/import/buy

> 購入 - purchase  
>  輸入 - import  
>  買う - buy  
>

for cars

> 新車 - new car  
>  外車 - import car  
>
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Here are some more verbs that are related to having, receiving, and owning.

These ones are common:

くれる - to give (usually to speaker)  
もらう - to receive (from somebody)  
ゲットする - to get, to obtain  
受{う}ける - to get, to receive  
得{え}る - to acquire, to get  
取{と}る - to take  
含{ふく}む to have, to include  
抱{かか}える - to hold/carry in/under the arms, to have (problems, debts, etc.)  

There ones may be more rare or formal:

達{たっ}する - to reach, to get to  
擁{よう}する - to have, to possess  
有{ゆう}する - to own, to have  
所有{しょゆう}する - to own, to possess (property)

Also, 持つ has an advantage over ある because you can leave out the topic/subject
without changing the meaning.

> 車がある。 There is a car.  
>  私は車がある。 I have a car. (As for me, there is a car.)  
>  （私は）車を持ってる。 (I) have a car.
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2015-11-18T20:35:22.880
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Like Saitama and Darcinon mentioned, there are several expressions you can use
for different situations here, depending on the context. I don't think there's
a universal answer here though.

If you want to ask "do you have/own a car?" The most natural way of asking
would be

> 車を持っている？

In theory you could use 持っている with the game character as well, but in that
context these terms might be better.

> 大乱闘スマッシュブラザーズでワリオをゲットした？  
>  or  
>  もうワリオを手に入れた？
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> you're talking to a friend on the phone... if I asked you if you had a car.
> Would I just say あなたは車がある？

I would say 「[車]{くるま} [持]{も}ってる？」 or 「車 ある？」

> What about a new character in a game? How would I say 'do you own him' or
> 'have you got/received him yet'?

「XX手{て}に入{い}れた?」 would be fine, but to sound more casual I would say 「XX
[取]{と}った？」 「XX [取]{と}れた？」 「XX ゲットした?」 or maybe 「XX もう持ってる？」

> You want to ask if they have a new game

I would say 「XX (もう)買った？」 or 「XX もう持ってる？」
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