Baptism is for the child yes but infant baptism demands that the Parents and God Parents profess their faith. The Parents are the first teachers of the child in the faith, therefore one of the requirements for baptism into the Catholic Church is that there is the hope that the child will be raised in the Catholic faith. One of the sponsors also has to have received the sacraments of initiation (Baptism, Eucharist, and Confirmation).
Only if the baptism is to be perfmmred by a Catholic priest in a Catholic church
but any Christian baptism is a Christian baptism, per Catholic practice. Personal experience.