I don't understand why the author is so fixated on the idea that "a business owns nothing for itself." The statement doesn't seem to mean anything, its just a confusion about what equity is. Equity is the value of the portion of the assets that are not owed to creditors (not liabilities); in some structures, the accounting of equity is divided between the multiple owners of the entity, but not always.
Double-entry bookkeeping is just this obvious fact, taken to a conclusion: every transactions of money is balanced; an equal amount of money is credited and debited in each transaction.
When I teach basic accounting to colleagues, I usually start with the accounting equation:
Capital = Assets - Liabilities
Then later I go on to explain that you can rearrange that equation:
Assets = Capital + Liabilities
One way to think of that is that the sum total of everything the business has (Assets) has a claim on it from either creditors (Liabilities) or owners (Capital).
So, it's really true that the company owns nothing for itself.
A lot of people have difficulty getting their heads around double-entry and, therefore, accounting in general. Different explanations seem to work for different people.
For anyone looking for an intro to accounting, I highly recommend Frank Wood's "Business Accounting 1".
Yes, it's a terribly confused explanation of the concept 'equity'. For example it has no way of explaining the difference between the paid-in amount and par value of stock, as far as I can tell (is there a word in English for this difference, btw? In Dutch it's called 'agio' but I can't seem to find one specific word for it in English).
Agio is used in English as well. It is an Italian word denoting the difference of two numbers due to a diverging method of assessing it. It is thus most often used to denote the money banks keep in various transactions. The meaning you specify in Dutch is most closely related to the use of agio to name the difference of the face value of silver coinage and their real metal content.
I think equity is the hardest concept in accounting for people to intuit. It's easy to understand what Assets and Liabilities are, and what Revenues and Expenses are. It's not even that hard to understand accrual concepts of depreciations.
Double-entry bookkeeping is just this obvious fact, taken to a conclusion: every transactions of money is balanced; an equal amount of money is credited and debited in each transaction.