One of my Indian coworkers said you could tell a lot about someone from their last name. Family origin, for one, down to a pretty fine granularity. Caste, likewise. My closest Indian friend once asked me if I could guess his caste. I told him I wouldn't even try. He laughed and told me that any Indian would instantly recognize his last name as placing him not only as a Brahmin but at a specific position within caste. In fact he hides it a bit, going by last initial when he can. I think you're right that a lot of Indians would be glad to get out from under their last names.
Around independence many Indian families kind of chose their last names (or had a choice imposed on them, in the case of lower caste members). Surnames were kind of a European import and traditionally people were identified by a personal name and one of several specifying markers like their home village, their familial vocation, religion, or jati. The Jati-based names are the ones people are talking about as being identifying. People who weren't enthusiastic about their caste identity tended not to pick identifying names and would have named themselves after their hometown or something instead. But even that has some associations.
A jati is what people talk about when they talk about 'caste.' They're sort of like large, extended kinship groups analogous to Clans in Scotland. These groups all have associated vocations akin to how in the West we have stereotypes like linking Jews with banking or Koreans with dry cleaning. There's all sorts of legal, practical, and historical reasons why these associations take hold. And since they have associated vocations, they have associated Socio-economic-status markers as well. It's not all that different from ethnic discrimination except more concentrated and formalized since it has a much longer history.
Untouchability is kind of a special category. Those groups were sort of pre-designated as being outside the pale of polite society. This was largely informal and variegated around India but become codified around the late Gupta empire (around 500-600AD). There was still a sort of group or ethnicity-based form of social mobility around this time. My own jati, for example, would have been considered among the lowest of the low in antiquity but by the time of the British Raj they had became more of a "middle-tier" caste. This was much more common in South India than in the North where the lines were never very formalized or rigid. There are all sorts of records of geneologies being created to link caste groups to mythological heroes to raise their profiles.
The British Raj--by creating a sub-continent-wide government, a census, and a literal-to-the-letter interpretation of Hindu law books that exaggerated the traditional influence and role of priestly castes--ended up hardening and ossifying these pre-existing cultural mores into hard rules that we're living with to this day. You notice a big difference in older diasporic Hindu communities, such as in the West Indies where Indians were brought over as bonded labor. Most high-caste brahmins absolutely did not go because of a superstition around losing your caste if you cross the ocean. BUT they still observe many of the same rituals and religious rites. It turns out there were low-caste/low-status mendicant priests around. They just weren't the high-status ones that the Raj and Western anthropologists talked to when they started defining what "Hinduism" is.
