Tuva was annexed by Tsarist government in 1913 IIRC. Only ROC has this claim because PRC recognized Outer Mongolia.
After reading a little bit on this issue, the Chinese claim looks extremely weak to me. They have never really ruled the area, it was always very autonomous and free of Chinese troops and administration. Even calling it a protectorate would be a stretch.
That's where the Qing-Russia border was in 1912, then Russia pulled a Putin when China was having a revolution.
And Taiwan does recognize Mongolia officially, see the map by their MoFA:
http://www.mofa.gov.tw/CountryAreaI...BE749&n=A985E71D2A3FA4B6&sms=26470E539B6FA395
Just that they don't want to go through the trouble and accidentally pissing off China when doing so.