A handle or grip — the part of an object designed to be held (e.g., on a bag, pot, or mug).
鍋の持ち手が熱くて火ばさみが必要だった。
The pot's handle was hot, so I needed tongs.
このバッグは持ち手が丈夫で長持ちしそうだ。
This bag's handles are sturdy and look like they'll last.
コップの持ち手が小さくて指が入りにくい。
The mug's handle is small, making it hard to fit my fingers.
A holder — a person who holds something (literal or abstract), such as authority, a position, or an object.
このプロジェクトの資金の持ち手はどこですか？
Who holds/funds this project's money?
本当の権限の持ち手は部署長ではなく、その委員会だった。
The real holder of authority wasn't the department head but the committee.