The 21-year-old, who made the step up from Swansea City over the summer, has hit the ground running with the Red Devils. James is the club's top goalscorer this season with three goals from four appearances in the Premier League.
Manager Ole Gunnar Solskjaer was keen to bring in young British talent over the summer, with James and Aaron Wan-Bissaka coming through the doors at Old Trafford.
With James now linking up with the rest of the Wales squad ahead of their Euro 2020 clash with Azerbaijan on Friday night, Giggs has been pleased with how the winger has started with the Red Devils, but not surprised.
"It doesn't matter where you are in the world or what kind of ability you have, confidence is everything, confidence builds," Giggs said.
"We all know about what he is capable of, what he's done especially over the last eight or nine months. He just gets better and better.
"He is at the right club. Ole will look after him in the right way, he'll know when to play him. So no, I am not surprised. It is a club that loves wingers, they love players running at defenders and exciting the fans.
"I'm pleased for him, he won't get carried away, he's that sort of character. He's really down to earth and he is at the right club in regards to keeping his feet on the ground."
James will be looking to extend his goalscoring run when Manchester United return to Premier League action on September 14 when they host Leicester City at Old Trafford.
The Foxes, who have impressed under manager Brendan Rodgers, will take their unbeaten record to the Theatre of Dreams after a strong start to the season.