This Olympics sees a British team competing at a Games for the first time since 1960 but only after officials from Scotland, Wales and Northern Ireland objected on the grounds that their status as independent football nations would be compromised by such a side.
Manchester United great Giggs was named captain of the team by manager Stuart Pearce, the former England defender.
Welshman Giggs will be competing at the first major international tournament of a medal-laden career -- Wales's last appearance at a leading global football event was the 1958 World Cup.
Giggs, one of only five Welshmen, including Liverpool forward Craig Bellamy in an 18-strong squad featuring 13 Englishmen, said Monday: "I hope it is not a one off, based on what I have experienced and how much I have enjoyed it this week, I hope in future Olympics there will be more Great Britain teams.
"Both Craig and I are Welsh and proud of it, but for me it was a chance I could not turn down, of playing in the biggest sporting event in the world," the 38-year-old Giggs added.
"I am glad I have done because so far everything has gone great and hopefully that will continue.
"I have had nothing but positive feedback from the people I know, wanting me to do well.
"The different FAs may have opposed it, but people who I have met face to face have been nothing but positive.
"I am not a politician, but from what I have experienced, I just hope fellow British footballers can experience what I have."
Liverpool forward Bellamy -- one of Pearce's three overage picks, which did not include former England captain David Beckham -- endorsed Giggs's comments by saying: "It has been immense. The week away could not have gone any better, all the boys have really got on.
"I never thought this was going to happen, for me to be involved in anything like this. I just want to embrace this and enjoy every moment. This is something I am going to cherish for the rest of my life."
Meanwhile former Manchester City manager Pearce was in confident mood ahead of Friday's friendly with gold medal favourites Brazil in Middlesbrough, north-east England.
"As a group we have gelled fantastically well, I am really pleased with the way they have come together, which has been quicker than I thought," said Pearce. "The venue in Spain was ideal and we are delighted with the week we had.
"I set teams up to try to win the tournament with the best players I have available.
"A partial success as well would be all of these players going back to their clubs saying what a fantastic experience they had."
Chelsea striker Daniel Sturridge has now joined up with the squad after a spell in hospital with viral meningitis.
The 22-year-old will be assessed as the squad head north to prepare for the Brazil game.
"There was certainly doubts because of the illness he had," said Pearce.
"Our fitness and conditioning people are having a look at him, to see how he will come through training.
"Chelsea's medical people are more than happy, but I have got to be 100 percent sure in my mind that Daniel is fit enough and can see out a tournament."