The young artist hopes to make his mark with his unique style.

make one's mark 是固定短语，意思是"留下印记、取得成就"，常用来形容某人通过努力获得认可或成功。例句中 his 替代了原短语中 one/your 的位置，指代"他的"，使短语可随主语人称灵活变化。