Properly "Big X" refers to the biggest players in the X industry, particularly with regard to trade groups and lobbying. It's useful because often this section of the industry will have unique qualities compared to the industry as a whole or other industries (such as how much they spend on lobbying, how much innovation they are responsible for, etc)
It is true that people will often use it as a lazy and fallacious way of vilifying but please note this is not exclusively a progressive phenomenon. Many of these groups lobby for things like govt subsidies, laws which restrict personal freedoms in ways that help their industry, etc...many conservatives aren't thrilled with such behavior.
In a brief non scientific survey of a few subreddits (r/libertarian, r/socialism, etc) phrases like "big oil" appear roughly comparably.
People on both ends of the political spectrum are (perhaps rightly) displeased with some of the actions of such groups and vilify them - they just disagree about what the root cause is and how to prevent such behavior.
The fallacy comes in when people use some particular actions of some players in an industry to invalidate everything that comes out of it. (such as implying that anything produced by a startup is ipso facto healthier than something produced by a large pharmaceutical company)