The basic reason is that most Indians do not have family names . Faced with having to come up with a last name, Indians have resorted to various means to get last names. Some of these are so common that most people do not even realize that they have had to , in effect, discover their last names ...
>I am sure a lot of you had to explain to your western >counterparts why your first and last names are switched. >It seems only Tamils do this. The North Indians, the Telugus, >and the Sinhalese use a last name. Any reason why the Tamils >do not use a last name? Did they ever use a last name? Is it >a western tradition only?
Many Tamils use a "vilasam". That gives the initials (and in Tamil, the first letter is actually a syllable) of all your paternal ancestors upto, say, 7 generations. This keeps every one readily identifiable. For e.g., in a reasonably-sized community,
Mu.Ko.Ka.Mu.Tha.Er. Ganesh would be the cousin of Mu,Ko.Ka.Mu.Tha.Ka. Ganesh.
Nowadays, the complete "vilasam" is used only in solemn occasions -- more commonly, only the first initial (father's) is used. As for why we don't use a last name, my guess is that a last name like a caste name is really not identification enough.