Many protestants consider themselves saints. One definition of saint, according to Mirian Webster, is one of God's chosen and usually Christian people.
Often in new testament letters, Paul and other writers address the group of believers as saints.
According to the catechism of the Catholic church, the church is the holy people of God, and her members are called 'saints'.
And protestants are covenant people as well; the new covenant that Jesus spoke of.
Many protestants also consider themselves priests as well, and would disagree with your assertion about proper priesthood authority. "You also, as living stones, are being built up a spiritual house, a holy priesthood"
So, LDS are not the only ones referring to themselves as a "saint" or "priest".