This is absolutely incorrect. Saturday is called "Shani"-var (pronounced shunny) after the name for the Hindu god of justice [1]. Hindus also map his presence to the planet Saturn. Either way, no relation to the name Sunny at all. Source: several Indian friends of ages 18-50.
More commonly Shani is the harbinger of bad luck & retribution, which is why you’ll rarely see any Indian origin person named Shani. In a Hindu mythology Shani is the elder brother of Yama, the lord of death & justice.
Sometime people named Sunil, such as the cricketer Sunil Gavaskar, use Sunny as a nickname.
Hey, maybe it factored into the decision to name them, although it's unlikely.
From what I understand, the reasons for picking nicknames are quite random. As another commenter mentioned, people named Sunil or Sanjeev go by Sunny sometimes, although other times it is just picked out of the blue, like with Bollywood actor Sunny Deol, whose real name is Ajay. His brother Vijay uses the screen name Bobby, by the way. Random.
1: https://en.wikipedia.org/wiki/Shani