They are about the only one who can call him by a nickname.
"彼をニックネームで呼べるのはアイツくらいだ"
- They are about the only one who can call him by a nickname.
- They are the only one who can use his nickname. （ニックネームを使えるのはアイツだけ）
- The only one who can call him by a nickname is them.
ここでは "they" を使いましたが、"he" か "she" を使っても大丈夫です。
"Them" -> "her" か "him" （それか名前を使う）
He's the only one who can call him by his nickname.
There's no one else who can call him by his nickname other than him.
ーHe's the only one who can call him by his nickname.
「彼のことをニックネームで呼べるのはあいつだけだ」
He's the only one で「彼は唯一の人です」
to call him by his nickname で「彼のことをニックネームで呼ぶ」
ーThere's no one else who can call him by his nickname other than him.
「彼のことをニックネームで呼べるのはあいつ以外誰もいない」
There's no one else で「他に誰もいない」
other than him で「彼以外」
ご参考まで！
He's the only one who can call him by his nickname.
He's the only one who can call him by his nickname.
「彼をニックネームで呼べるのはアイツくらいだ」
関連用語
nickname: ニックネーム、愛称
only one: 唯一の人
respect: 尊敬
この表現で、その特別な関係と信頼を伝えることができます。参考にしてください。
アメリカ合衆国 
カナダ 
