To tame an animal — to train or make it docile and accustomed to people.
彼は野生の馬を時間をかけて手なずけた。
He spent time taming the wild horse.
子犬を手なずけるには根気が必要だ。
It takes patience to tame a puppy.
To win someone over — to ingratiate oneself or manipulate someone into becoming friendly or compliant.
彼は上司を手なずけて重要なプロジェクトに参加させた。
He won over his boss and got him to join the important project.
営業担当はクライアントを手なずけるのがうまい。
The salesperson is good at winning over clients.