Interesting comment. In common parlance in the past, "two sets of books" always meant on set that was real and reflected what happened....the second set was for government taxing and regulators...which was often false and showed different results.
In a real and perfectly legal manner, corporations (and other entities) may have two sets of books. One based on financial statement and standards as prescribed by the AICPA, and the other set of books may be kept as allowed by either a regulatory agency or for tax purposes. For example, regulated utilities might have to maintain records in accordance with the agency regulations and yet report financial statements that show different results to say the SEC. Federal Tax Regulations might allow for a third set of "books" or financial statements. There would be reconciling workpapers between the statements in the background.
A very common situation for small corporations is that for financial reporting, they use the accrual basis of accounting...but for tax purposes, they would use the cash basis of accounting. Both legal, but in any given time period would show different net/taxable income.