This use is countable: a Tamil, two Tamils, many Tamils. It names a people, not a country, so a Tamil may hold any nationality, and Tamil origins are enough. Large Tamil communities live outside those two regions, which is why you meet the word in writing about Singapore and Malaysia as well. The language sense of the same word is uncountable, and that difference in grammar is usually what tells you which one you are reading.

It is a Dravidian language, which means it belongs to a family of its own and is not related to English. As a language name it is uncountable: you say she speaks Tamil, never a Tamil or two Tamils. The same word also names the writing system used for it, as in Tamil letters. For the people themselves, the word is countable, and that is a separate entry.