Nope. Mormonism is not part of either Protestantism or Catholicism. We actually see Mormonism as being identical to the original church Christ established while on earth, with prophets and apostles, revelation, priesthood authority, temples, etc. We believe that after Christ, all the apostles were killed off, the priesthood authority was lost, and the church ceased to exist. Many gospel truths were preserved, and people tried to live the gospel, but they lacked authority to administer baptism or other ordinances, and they didn’t have direct communication with God.
We believe it was necessary for there to be a “restoration” of the true church, which could only be accomplished by messengers from God restoring the priesthood authority, since nobody on earth had it and there are certain required methods for granting it.
In other words, we see all Christian religions as being outgrowths of Mormonism, which is the true and original Christian church.