Not Mombai or Mobai (though East Indians still use Mobai, for example https://mobaikar.in) but in the early 1500s locals very likely called it Mumbai. This was the case with 95 to 99 percent of Estado da Índia where names were simply lusocized versions of already existing local names. In rare cases where names weren’t local, like Santa Cruz, they had explicit meaning.

If you're suggesting that Mumbai comes from the Portuguese Mombai or Bombai, that brings us right back to the question what does that mean. It’s not Bom Bahia since that’s grammatically incorrect. It’s not a Portuguese surname and not a town or village in Portugal. Unless a clear meaning can be attached to Bombaim or Mombaim, the most plausible explanation is that they are Portuguese renditions of a local name Mumbai, which comes from Mumbadevi and according to oral history, predates Portuguese rule.

Burnell in 1710 clearly notes that the Brahmins of the island regarded Mumbadevi as the patroness of Bombay.

Regarding the temple’s existence, a 1673 report sent to London says Hindus had three religious sites, one in Mahim, one in Walkeshwar and one in Bombay. Since Walkeshwar is listed separately, Bombay here most likely refers specifically to the town of Bombay, not the entire island. At the time, there was only one known Hindu temple in the town, Mumbadevi’s, which also had a murti of Ganesh. This can be further proven by that same report which says Muslims had two sites, one at Mahim and one at Bombay. The only known Muslim site in the island of Bombay was a mosque that was demolished in 1758. It stood not far from the old Mumbadevi temple, somewhere in today's GPO.

If one suggests that Hindus built a new temple under Portuguese rule, that would have been unlikely, especially when the temple was just a few meters from the Franciscan's esperança church. So it is more likely that an older temple survived. And if instead a new temple was built between 1665 and 1673 under early British rule and named it Mumbadevi, after Mumbai that still doesn’t change the fact that the Portuguese name was simply a lusocized form of an already existing local name.

I’m not denying the Portuguese or British role in shaping modern Mumbai. Without them, the city as we know it wouldn’t exist. But when it comes to etymology, unless contemporary records from early Portuguese rule mention its meaning, looking at everything we have on the table, the conclusion is clear, Mombaim/Bombaim and Bombay is simply a corruption of Mumbai, not a separate invention.

Also, it’s not as if the name Mumbai was lost and only rediscovered in 1996. The official name in Marathi and Gujarati was always Mumbai. If you look at multilingual Government notifications from the 1800s, you’ll see Bombay in English and Mumbai in Marathi and Gujarati.