Since some Indian family names can be used to guess the family's region of origin:
A person whose family name is Gaikwad and who is probably from Mumbai, would he be likely to have Hindi as his first language?

There's at least one Gaikwad who writes in Marathi, the principal language of Maharashtra, the state of which Mumbai is the capital.

Yes, that site isn't well done; it seems to assume that you've got there by following the Marathi authors link and don't need to be told what language he writes in. But if you click on Marathi in the list of languages at the bottom, you go back to the Marathi authors site and Mr Gaikwad is there.

Looking for something else in my hard copy of Hobson-Jobson I see that gaekwad or gaekwar originally meant cowherd and came to be the dynastic name/title of the Maharaja-Gaekwar of Baroda.
I wouldn't have come across that if I'd been using the online version.
The etymology reminds me a bit of the house of Stuart, etymologically identical to sty-ward, keeper of the sty, although probably sty or stig in their name meant more like household than pigsty.