The total is one measure of U.S. wealth. In a way, it includes double-counting. For example, the total includes values of mutual fund shares, and of corporate shares owned by mutual funds. But that is similar to pointing out that if we all tried to take our money out of banks, we would find it's not all there. So long as our financial system doesn't collapse, the total can be viewed as one measure of U.S. wealth.
Financial institutions own almost all debt.
When the promise to pay is signed by a government or a big corporation, its value is clear and it is easily bought and sold. Financial institutions buy and sell that kind of debt by telephone. That is the bond market. Households own about two-thirds of all equity. Equity entitles the owner to a share of profits, and to a share of proceeds if the business is sold.