Taiwan might be a part of "China" but the CCP has never controlled Taiwan. So they can consider it a "breakaway province" but from the Taiwanese prospective, they were never controlled by the CCP. If they want to bring the Emperor back then maybe they have a point. You may as well consider the mainland as a breakaway province.
The only analogy I can make is if say, the CSA instead of surrendering fled to Cuba, and the USA could not finish the job because the British navy was too strong. I couldn't really justify invading Cuba on some historical claim.