Business can sometimes mean company or firm. However, can it be used in the way company or firm are used?
For example, can I say:-
"He is the CEO of the business."
"It's a TV business."
"A business dealing with drugs."
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Answer: Yes!! (Not always though..)
Let's consider your examples again,
Acceptable- "He is the CEO of the (biggest) business (in the history)."
Incorrect- "He is the CEO of (the/our) business."
Acceptable- "It's a TV (making) business."
Incorrect- "It's a TV business."
Informally acceptable- "A business dealing with drugs." (Should be "in" instead or "with".)
So, as Barrie already pointed out, the usage usually depends on what came before and what follows the sentence. The word 'business' doesn't necessarily always mean a company.