Is there a word for fake company history?

Solution 1:

The term backstory (or back story) is used in literature to describe the fictional bases of the plot and characters.

If the company began to believe their fiction, it might be called corporate confabulation

Solution 2:

Such a history might be referred to as a humbug (“something designed to deceive and mislead”). Humbug also means “a willfully false, deceptive, or insincere person”, so a company with a humbug history can be called a humbug as well. Postiche (“an imitation, counterfeit, or substitute” and “anything that is false; sham or pretence”) also might do, as might ruse.

More formally, the made-up history is false credentials. False can be replaced by a synonym or by related words such as sham (“Intended to deceive; false”) and bogus (“Counterfeit or fake; not genuine”).

The history can be called a cover story (“a fictitious account that is intended to hide one's real motive”) if the company is a front (“A person or institution acting as the public face of some other, covert group. [eg] Officially it's a dry-cleaning shop, but everyone knows it's front for the mafia”) for smugglers, spies, or criminals.