It's definetly true!!
The Mumbai consulate requires you to change your name on your passport. I recently got married, and I was hoping that my wife would come to US with me.. However when we went to Mumbai consulate, they said that they wont process her H4 until she gets her name changed to her married name. So, I know it's definetly true for Mumbai consulate because I had to stay away from my wife because of the stupid rule.
Note that name change in Mumbai takes only a couple of days, as long as only name change is required. So,l make sure your fiancee gets her address and other info updated before you go to India
However, there is some confusion with the way the Mumbai Consulate words it requirements. I dont have the exact words, but it sounds like if the dependent is male, the he will have to take his wife's name!!! So, there was a rumor going around that Mumbai consulate is making guy's adopt their wife's surnames, just because the wife is on an H1!! This is even more absurd that making the wife change her passport!! and thankfully, not true.
There was some clarification posted somewhere later, that if you are male and you are a dependent, then you need to include your spouse's name in your passport, not change your last name
