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The Duke of York, 46, revealed details of family life during his childhood, including how he used to watch evening television with the queen, rollerblade and race miniature cars around state apartments with his younger brother Prince Edward.
She was in touch with modern trends through her grandchildren, said the prince, fourth in line to the throne.
The Royal Navy veteran said the decision of two of them, Princes William and Harry, his older brother Prince Charles's children, to join the British Army proved how close the queen was to her subjects because of the risks they could face in combat for their country.
"The queen is a great mother and a great parent, and so too is the Duke of Edinburgh (her husband Prince Philip), and we used to have wonderful fun racing up and down here at weekends, we'd do all sorts of things," Andrew told Sky News television.
He recalled a childhood scrape when Queen Elizabeth helped him up after he came off his tricycle at Windsor Castle, west of London.
"Yes, she used to pick us up and she'd say "don't be so silly, there's nothing wrong with you, go and wash it off". Like any parent, there's nothing wrong with you, go and clean up."
Television with the children
Queen Elizabeth has combined being head of state in 16 countries with being a mother of four (Charles, Princess Anne, Andrew and Edward) and a grandmother of seven.
Her third child described how the royal family used to sit together in front of the television to watch the BBC's sports program.
"It was usually at tea time and we used to watch 'Grandstand' on Saturdays and in the summer, gosh I can't remember what it was, it was a Sunday cricket league we always used to watch." He also recounted hiding behind the sofa during scary programs.
Prince Andrew said of the queen's image: "She is seen as consistent, she's always been there, she's always there and a part of the United Kingdom and that's what's so enduring about her.
"She has a relationship with Britain, she has a relationship with every single individual person in the United Kingdom. I just have a different one because she happens to be my mother at the same time but I still see both sides of the same coin if you see the analogy."
Andrew said that the queen kept up to date with modern youth through his nephews William, 23, Harry, 21, and his own daughters, Princesses Beatrice, 18 and Eugenie, 16.
"Life is a continuous change and I think that the art that the queen has achieved is ability either to remain concurrent with the change as it happens or lead that change.
"I mean for goodness sake, she has got grandchildren who are up to the minute in what's going on out there so she has the knowledge, she gets it from not only my children but she also gets it from William and Harry, so she's up to the hunt, she knows exactly what's going on."
Andrew, Britain's Special Representative for International Trade and
Investment, said the queen had given him coaching during foreign tours on how to conduct himself.
"There were occasions. Stand here, do that or don't do that," he said.
Military connection
The prince, who put himself in the firing line as a helicopter pilot during the 1982 Falklands War between Britain and Argentina, discussed the possibility of William and Harry serving in a conflict zone.
Harry joined the Blues and Royals of the Household Cavalry regiment earlier in April as Second Lieutenant Wales. His elder brother, Officer Cadet Wales, is still training at the elite Sandhurst academy.
"I think with the modern way of the armed forces, it is inevitable that they are going to see action of some kind somewhere in the not too distant future, particularly as young officers, because we are already out in Iraq and Afghanistan.
"They are open to those sorts of deployments just as everybody else is, and open to the same risks as everybody else, as I was. I could have been sunk, I could have been shot down, you have to take your run with everybody else.
"That is another aspect of that relationship that the queen has with the country because the country then knows that children or grandchildren of hers are actually taking the same risks as everybody else and that's another reason why the queen is so close to people."