Note that it would likely be odd for someone to be just named Tashina, see the examples mentioned: Tashina is a part of a name, not a full name itself. According to Native Languages of the Americas, "Tashina (pronounced tah-shee-nah) is a popular Lakota Sioux woman's name." It seems this is now used as an independent name.
Tanisha was borne by a Hunkpapa Lakota Sioux woman who fought in the Battle of the Little Bighorn) and Tashina Topa Win ("Her Four Robes" or "Four Blankets Woman"; borne by the younger of Sitting Bull's last two wives). It is associated with traditional dancing. The name was brought to public attention by a Cheyenne character in the movie 'Windwalker' (1981), in which case it may be an alteration of the Cheyenne name Tashena, a shortened form of a longer name derived from taa'eeshe'he "moon".
According to Native Languages of the Americas, "Tashina (pronounced tah-shee-nah) is a popular Lakota Sioux woman's name." It seems this is now used as an independent name.