The practice amongst Hindus in modern India varies. Indian Passports are standard with First Name and Last Name fields.
In North India, for many people surnames have been standard for generations and are dependant on caste and family group. Some family groups may have multiple interchangeable surnames available to them, so that a child of someone called X LastName1, may choose to call themselves Y LastName2, where both LastName1 and LastName2 are indicative of the same family group.
These names can be derived from traditional job of the family, or the name of person the family is supposed to be descended from, or some feat performed by the celebrated ancestor or the place the person's ancestors are supposed to be from.
In many cases, for example in rural India, or among the lower social strata, or just generally where this historic information may not be available to the family, or because the person in question wants to change their surname for any reason - generic Last Names like Kumar, Singh (can be used by any gender) or Kumari (female LN, usually used by unmarried women/girls) and Devi (female LN, usually used by married/older women).
In the South Indian state of Tamil Nadu, there is no practice of surname - instead father's name is used (uncommonly mother's name for some girls, and after marriage husband's name). In countries outside of India with significant Tamil minorities this is implemented as male_person's_name S/O father's_name or female_person's_name D/O father's_name in official documents.
In Karnataka, also in South India, surnames are present - they may include some element of 'place of origin' and a patronymic as an 'extra' name. Keralite Hindus seem to have the tradition of using caste name as surname.
As for the people in Vedic literature not having surnames, well, surnames all over the world are a fairly recent innovation! People in ancient lit all over the world rarely seem to have them. Case in point, the Arthurian legends, Norse sagas etc. In fact in many countries, using patronymics as the Last Name is still a common tradition. :)