Yes, the Same Family as Actual Roses
This one tends to land sideways. The strawberry's botanical family is Rosaceae — the rose family. It is, in a very real taxonomic sense, a cousin of the rose growing in your neighbor's garden.
The rose family is large, and once you know to look for it, the resemblances start to make sense. Apples and pears are in there. So are peaches, plums, cherries, apricots, almonds, raspberries, and blackberries. They share a five-petaled flower structure, certain leaf patterns, and a way of producing fruit that involves the receptacle of the flower more than most other plant families do. The next time you see a strawberry blossom — small, white, five distinct petals — try mentally placing it next to a wild rose. You'll see it.
This kind of relatedness is one of the quietly delightful things about botany. You can be standing in front of a fruit, a tree, and a flower and have no idea that, evolutionarily speaking, you're looking at three relatives. The strawberry is closer kin to your rose bush than to a blueberry, which is in an entirely different family. Names are noisy. Family trees are not.