No. She is the sovereign, the head of state, and the head of the nation. The British royal family owns a certain amount of private property, but the largest collection of property the to which the sovereign is entitled is called the Crown Estates. These holdings were administered by the sovereign until the mid-18th century when they were turned over to the Treasury.

This was done in exchange for eliminating George III's debts and relieving him of responsibility for the cost of government. In exchange, the sovereign was paid an annual sum called the Civil List. There were some problems with expensive socks and embezzlement by clever butlers and the politically inconvenient granting of money, so they changed that system just a few years ago so that she gets a cut of the annual revenues from the Crown Estates.

She doesn't really "own" the British government and ships and such. When they say "Her Majesty's Ship" and "Her Majesty's Secret Service" it's all just symbolism and tradition and ceremony.