Is Tamil a language

Yes. It's a Dravidian language spoken mainly in southern India, Sri Lanka and Singapore.

And, The term "TAMIL" mentions not only the language, but also culture, civilization, tradition, music & etc., which are originated from Tamil people and Tamil soil.

E.g: Tamil Tradition, Tamil Civilization, Tamil Music. (These have got the name of Tamil not because of the language, but from the culture whre they originated)