Why is this salt used specifically in Fehling's solution, instead of a simple disodium or dipotassium salt of tartaric acid?
Fehlings soluition is alkaline. Mixing cupric ions with an alkaline solution would result in formation of cupric hydroxide of very low water solubility. By adding tartrate ions, the cupric ions will form a complex with the cupric ions which will keep them dissolved in the alkaline solution. The type of cation of the tartaric acid is of no importance other than for solubility reasons.
The same principle is used in Benedicts´s solution, where tartaric acid is replaced by citric acid or a salt thereoff.