Traditionally, most religious changed their names when they entered novitiate (not at vows). However, the American Trappists at Gethsemane got a new name as they entered. This is documented in Merton’s “Seven Storey Mountain.” Rumer Godden’s “In This House of Brede,” set in a pre-Vatican II British Benedictine Abbey for women, also had entrants get religious names upon entry, she based the book upon Stanbrook, so I would assume this used to be their practice.
The vast majority of women religious in the US do not take religious names anymore, though the more traditional communities continue the custom. I believe the same is true of men. Some, such as the Religious of the Sacred Heart (women) and Jesuits (men), never did.