Maybe the dada-dadi would want him to keep it cuz he's their waaris but he's Aarti's son, not theirs. The only reason for the marriage was so that Ansh could have a father so it would make sense for him to have his father's name no? Of course you can argue that Prashant's the father so why not his name but even then, Yash has been more of a father then Prashant ever was and had left while Aarti was still pregnant.
As for Ansh always knowing that Yash is not his father, that's true to an extent. Ansh will always know that Yash is not his biological father but a father nonetheless.