No, stock is a claim to the equity of an enterprise, i.e., the difference between and enterprise’s Assets and Liabilities. If a company has $100 million in equity and 1 million shares of stock, then each share claims $100 of that equity. If that company went out of business and sold its assets and paid its liabilities, then each shareholder would own a share of the $100 million equity in proportion to the number of shares he owned.
The USD is not a claim to anything, let alone the equity of the Federal Government. Even if citizens held shares in the Federal government they would be worthless. The Federal government has unfunded liabilities of over $200 trillion dollars.