By the time it is deemed a variant, it has been sequenced number of times across the country and the health authorities always have a code or name to refer to this particular strain.

They had a code but not a name. The media can't refer to things by a confusing jumble of letters and numbers, they need a name, and if there's no official name then the media will create one themselves. Hence "India variant".

This is basic PR, e.g. when Heartbleed was disclosed it was given a name so that people could discuss it and attach meaning to it.