so wheras in the North, you'd have Narayan Sharma's son being named Vishal Sharma, in Tamil Nadu Sriram Narayan's son would be Vishal Sriram.
So yes, the system does exist
Due Disclosure: South Indian, not a Brahmin, but I have many friends whose names follow this system.
Caste does not say your ancestors did that job, caste meant a person from a particular caste could only do that job and could not take up other jobs, people from other castes could not take their jobs and were disqualified because you were not born in a particular list of families. Some castes were prohibited from getting an education and were killed for this.
https://www.telegraphindia.com/opinion/what-s-in-a-name-ofte...
If someone doesn't want to share their caste, that's very easy to do.
Hence... discrimination amongst indians is usually around where in india you're from, what language do you speak (tamil, hindi, marathi, etc.), what tier college you went to (IIT?), obviously gender, and other factors.
Comparing Caste to race is completely wrong and it's a red herring.
Having every indian fill out what caste they are on their job application doesn't solve the problem.