I'm not sure. A more educated person could probably answer this in a more appropriate way.
Be careful with what I'm writing after this point, I might be full of shit.
"Media analysis" is a relatively recent concept (Google says it exists since 1930). If I'm not mistaken I THINK it replaced the old "Culture Studies" that tended to focus more on the high-arts, like Shakespeare, and it guided our eyes to understand how people shaped and were shaped by media of mass consumption, so, journals, TV shows, books, etc.. Paintings and music can be considered "art" and "culture", but what if I want a term that encompasses those things AND ALSO the news, scientific papers, letters? Those three things aren't "art". The term "media" covers all of these.
Consumption is an equally useful word. You can say you watch videos, but you don't watch music. If I want a generic term that encompasses all the ways humans interact, absorb, and express themselves around media, I'll use the generic term "consumption".
As to why people are using these terms more often? I have a couple theories.
Maybe it could be because people consume more types of media nowadays? If I want to understand what extremized granny into becoming a Trump supporter I want tools to help me analyze the memes, news and videos that she consumes. But also not always people had portable music, films, and libraries with them. Media consumption seems to be more constant and compact.
Maybe it could be because there's a bigger influx of people talking about media analysis on the internet now? At least it seems to be popular enough for every now and then there be a Twitter discourse about how the wrong people don't have media literacy while they do.
Maybe it could be that people aren't using these terms more often at all! it could be that we're just in the internet bubbles where people do.
I'm not entirely sure though.

