Againor, of course the sect that calls itself the “church of Christ” is Protestant. (And let me remind you that taking a name for a denomination does not give you that identity, this sect calls itself the "church of Christ, but is not the Church of Christ.)
One reason the cofC is a Protestant sect is that it teaches Protestant doctrine. Just the very idea that the church is built on the NT of the Bible alone (sola scriptura) is Protestant. The idea that anyone, anytime, anywhere can “restore” the original church back into existence just by picking up a Bible and using it as a “roadmap” or “blueprint” is a very Protestant idea.
The cofC did break away from another denomination, the Christian Church (Disciples of Christ) in 1906 (not the early 1800’s). The founders of the Christian Church (Stone and the Campbells) split away from the Non-Jurors Presbyterian Church Seceeding another Protestant sect. The N-J P C S broke away from the Church of Scotland another Protestant denomination, which schismed from the Calvinist church of Geneva. which broke away from the Catholic Church.
Today’s self-proclaimed “churches of Christ” is the result of a series of church schisms, all Protestant and eventually going back to the Catholic Church.
And BTW the so-called “one and only church of Christ” is still schisming into more and more, ever smaller denominations. I have counted at least seven Protestant denominations calling themselves the “churches of Christ”, all denying fellowship and the very name Christian to the others.