Naming conventions vary, and when you consider names across history/geography, it is the present-day Western convention of "GivenName FamilyName" that is unusual and needs explanation.
Generally speaking, someone is born and at some point days/months later, their parents start calling them by some name, while the rest of the world might also doing so at some point, possibly different people using different names. For purposes of interacting with administrative systems yet another name may be adopted. Only when it has been necessary to distinguish between multiple people with the same name do secondary names start getting used, either occupational descriptions (John the Baker vs John the Carpenter vs John the Smith) or places where they came from or were noted for (Jesus of Nazareth, William of Orange, Leonardo from Vinci), or disambiguating with parents' names (Mohammed bin [son of] Salman, Björk Guðmundsdóttir [daughter of Guðmund]) — these are all conventions still existing today, with occasional funny consequences when someone imagines one of these to be a "family name" that persists from father to child across generations. (See "what would Of Nazareth do" about people—even otherwise educated ones—treating “da Vinci” as such.)
Coming to India: there are different conventions. Typically just a name and an initial letter (placed either before or after the name) to distinguish between multiple people (in the same classroom say) with that name. When a boy was named "Anand" by his parents, because his father was "K. Viswanathan", he became "V. Anand" in school records, and this is the name I remember reading articles about this chess prodigy in Indian newspapers. At some point the international press started spelling out his first name and called him "Viswanathan Anand", putting his father's name first, and even started calling him "Viswanathan" or "Vishy" — he used to object and point out that they were calling him by his father's name, but eventually he just got used to it and even began to like it. In this generation, this boy was named "Gukesh" by his parents, and was "D. Gukesh" in school records and news reports, but somewhat wisely they decided for international sources to put the initial after the name, so "Gukesh D", and for those who cannot handle just an initial, spell it out to "Gukesh Dommaraju".
(You have had other replies claiming this to have something to do with Tamil Nadu anti-caste politics. While no doubt that movement discouraged the use of caste names as surnames, the initial convention pre-exists any of those political movements and exists in parallel in other states too. E.g. "S. Ramanujan" was the name on his early papers before the movement being spoken of. Some families/communities use surnames (in the sense you're thinking of) and some don't; that's all there is to it.)