In a sense, they do. To the extent one can describe an ecclesial body as Christian at all, it follows that it must hold to the basics about Christ. (This doesn't mean such a group is entirely orthodox, only that it isn't so heretical as to forfeit entirely the name Christian.)
Both Catholics and Protestants believe Christ to be true God and true man. Both believe Jesus to be the Second Person of the Trinity. And both believe in the Virgin Birth and bodily Resurrection of Christ. So both Catholics and Protestants can rightly be called Christians, though from the Catholic perspective Protestants are heterodox on a number of key points.
At the same time, there are many groups which style themselves Christian while denying basic christological tenets.
Jehovah's Witnesses, for instance, claim Jesus isn't Almighty God, Jehovah, but Michael the Archangel.
Mormons deny Jesus and the Father are the same God in two different Persons.
Christian Scientists, by rejecting the objective reality of the material world, repudiate the Incarnation as Christians traditionally have understood this doctrine.
There are also "liberal" theologians in the Catholic Church and within Protestant denominations who deny the divinity of Christ, his Virgin Birth, and his bodily Resurrection. As the late Hans Urs von Balthasar observed about a certain "liberal" Catholic theologian, when such basic Christological beliefs are abandoned, it's theologically incorrect to describe the one rejecting them as a Christian.
Obviously we Catholics can't forbid heterodox people from describing themselves as Christians, but we can make it clear they don't believe what Scripture and Tradition tell us about Christ.