The new United manager added: "I think it is a job everyone wants and not many have a chance to have and I have it."
Former Chelsea boss Mourinho succeeds Dutchman Louis van Gaal, who was sacked after United's FA Cup success in May.
The Portuguese said he was "frustrated" United are not playing in the Champions League next season but insisted he was at Old Trafford to win trophies.
United, the 20-time English champions, finished fifth in the Premier League last season, 15 points adrift of shock champions Leicester City.
They won the FA Cup, but the campaign was generally viewed as another disappointment, prompting the sacking of Van Gaal in May.
"It would be quite pragmatic to say let's work and let's try to be back to the Champions League, be back to the top four," said Mourinho.
"I am not good on that and I didn't want to be good on that. I am more aggressive. We want to win.
"For many years, success was just routine and in this moment the last three years are to forget. I want the players to forget. I don't want the players to think we have to do better and finish fourth. Finishing fourth is not the aim.
"I want everything. I want to win matches, I want to play well, I don't want to concede goals. I want the fans to be behind us.
"Of course, we're not going to get everything, but we want to."
Asked if he had a point to prove after his second stint at Chelsea ended in the sack, Mourinho responded: "There are some managers who haven't won titles for 10 years. Some of them never. The last time I won was a year ago."
Mourinho, who led the Blues to the Premier League title in 2015, added: "If I have something to prove, imagine the others. That's my feeling.
"I feel that I have to prove not to the others but to myself. I would never be able to work without success. That is my nature."
On new players
Mourinho said he had targeted four signings and had secured three of them.
"I am very clear in my approach and model to play and I need specialists in that," said the former Chelsea, Real Madrid and Inter Milan boss. "I like one or two multi-functional because you need them when you are in a bit of trouble."
Striker Zlatan Ibrahimovic and Ivory Coast defender Eric Bailly are already at the club, while Henrikh Mkhitaryan looks like being the third recruit after United's bid for the midfielder was accepted by Borussia Dortmund.
France and Juventus midfielder Paul Pogba, 23, has also been heavily linked with a return to the club he left in 2011.
"We decided four targets," said Mourinho, 53. "Until we have the fourth, we are still working hard. The third player will be official soon."
On promoting youth
Mourinho has been accused of putting his faith in proven, older talent at his previous clubs rather than investing in youth.
But the former Porto, Inter Milan and Real Madrid boss was quick to reject such allegations when he faced the media on Tuesday.
"You know how many young players I promoted to the first team from academies?" he asked. "Forty-nine. Some of them are big names, they are Champions League winners in the Euros, playing for national teams.
"One lie repeated many times sometimes looks true, but it will never be true."
On Ryan Giggs
Giggs, who made a record 963 appearances for the Red Devils, ended his 29-year association with the club last week.
The Welshman was assistant to Van Gaal and had been tipped by some to succeed the Dutchman.
"It is not my responsibility Ryan is not in the club," said Mourinho. "The job Ryan wanted is the job the club decided to give me. If, one day, he wants to be back to the club while I am here, I would never stop him."
On Wayne Rooney
Rooney has played the bulk of his career in attack and is only four goals short of Sir Bobby Charlton's club record of 249.
But he spent the end of last season in midfield, and played in the same position for England at Euro 2016.
"Maybe he is not a striker any more, maybe he is not a number nine any more, but with me he will never be a number six or somebody who plays 50 metres away from the goal," said Mourinho.
"Yes, his pass is amazing, but my pass is also amazing without pressure. To put the ball in the net is the most difficult thing you have to find."