These are, of course, net numbers. It isn't to say that the private sector isn't itself
split up into smaller sectors, stock, bonds, consumer credit, savings accounts, mortgages, whatever. That is what bothers me. But, I guess we have to go through this first.
Excellent graph. You can see the 100% correlation.
Right. We're talking basic stock and flow, that's about it, in terms of sectoral macro-accounting.
We can divide this up into the domestic private sector, domestic government sector and the foreign sector (the rest of the world, including firms, households and national governments).
Domestic private sector + Domestic government sector + Foreign Sector = 0
For example, let's say, for the sake of argument, that the foreign sector is running a balanced budget. The identity I used has a foreign sector balance of zero.

Let's also say that the domestic private sector's income is $500 billion and it's spending is $400 billion - a total net budget surplus of $100 billion for the year. By the identity alone, the domestic government sector will have a budget deficit of $100 billion over the course of the year. The domestic private sector will accrue $100 billion of net financial wealth, which consists of $100 billion in liabilities of the government sector over the course of the year.
If we use a second example, let's say that the foreign sector is spending less than its net income, with a total budget surplus of $300 billion. Simultaneously, during the same year, the domestic government sector also runs a budget surplus of $200 billion. When we get back to our identity, we know that the domestic private sector must have a budget deficit equal to $500 billion ($300 billion + $200 billion). During the same time period, its net financial wealth will have decreased by $500 billion as it must issue debt and sell off financial assets. The domestic government sector will increase its net financial wealth by $200 billion, decreasing its debt load and increasing its claims on the other sectors.
The bottom line is, if one sector runs a budget surplus, another sector must run a budget deficit. We're talking stock variables at the end of the day: one sector accrues net financial wealth, another must increase its debt load down to the last penny. It's impossible for every sector to simultaneously accrue financial wealth through budget surpluses.