To answer your question directly: Tuva was a component of Russia, not the Soviet Union.

To give a little more explanation: Blame Salchak Toka. He was a member of the Communist Party in the 1920s, and when a former monk, Donduk Kuular, declared Tuva independent from the Russian Empire, Joseph Stalin had Toka arrest Kuular. Toka was installed as leader of the new republic instead, and a couple of years later, in 1932, Kuular was executed.

Toka brought in fellow Communist Tuvans to Sovietize the republic. Farming was collectivized, Cyrillic script was adopted, local religions were suppressed, etc. In 1944 the Tuvan parliament voted to give up their independence and join Russia. Would the people have voted for this if there had been a referendum? Probably, as by this point they were pretty much dependent on the Soviet Union, and many Tuvans had joined the Soviet army during WWII.

The Tuvan people are proud of their language and culture, and since the fall of the USSR they have been electing more Tuvans to office - but they have generally not sought to leave the Russian Federation. Geographically, they are surrounded by very large, very dominant states - they can lean toward Russia or toward China for support, and they're sticking with Russia.

Note that the word "Tannu" hasn't been part of the name since the 1920s. If you're looking for more information on Tuvan history, this article in Soviet Studies is a good one, dating from 1992.