If Wikipedia is to be believed, godi media was coined by the TV journalist Ravish Kumar:
The term was coined and popularized by NDTV journalist Ravish Kumar, in reference to the sensationalist and biased Indian print and TV news media supporting the currently ruling NDA government. The term literally translates to "media sitting on the lap". In one of his shows, Kumar used silent actors to mime "godi media". This was accompanied by miming what the currently ruling party leaders wanted to listen to, using the Hindi film song "Bago Mein Bahar Hai".
(This snippet seems to agree)
It's hard to find the exact episode, but Kumar uses the term again in a 2017 interview.
It seems to have become a more popular term in around 2018, but there are much earlier uses of it. One of the earlier instances I was able to find is from July 2015, in an article from The Frustrated Indian:
We were literally moved to tears once we got to read this news. Upon careful examination we came across some startling facts. We realized that Rajiv Gandhi was the pioneer of many a revolutions in India and the ‘godi‘ media shunned them all. But we believe in giving credit where it is due. Rajiv Gandhi, the pioneer of Digital India Programme was also the father of the following revolutions.
I doubt it's used outside of India given that godi (गोदी) is a Hindi word, so unless you know Hindi, it's hard to understand the expression's meaning intuitively.
In the West, an equivalent would probably be lapdog media, distinguished from watchdog media.