I've just found a good article about this topic. I hope it answers to your question.
I got my twitter handle by asking.
One of my brand names (and my artist name) is Yuko Rabbit, and @yukorabbit was already taken.
So I politely asked the owner and explained honestly why I needed it. She kindly changed her username for me, and I could get my desired twitter handle.
1) Ask nicely as @HiYukoIm did
2) Offer to buy it
3) Shady, but if the account is old and unused you could spam report it with multiple accounts, hoping for it to get banned by twitter. Then you could re-register it. But that's theoretical. Never done it myself.
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