The reason why ordinary people parrot it is because that's what it was designed for. The proper term for "CCP" is "China." Referring to the Chinese government as the "CCP" (or the CPC) is like referring to the US government as the "Demoplicans" (or the Democrats and Republicans.)
Instead, we just say "the US government" or "the US administration."
CCP is a direct transliteration of the characters, so that's what it started as. Some time later China decided to change it but that's a lot of cultural inertia to move in a different direction.
I've never heard it called anything other than the CCP.
Unless the Communist Party of the US (I’m not looking up its official name, because it doesn’t matter) wins the next presidential election it’s unlikely that people will call it anything but the CCP. Everyone know what everyone else means.