18-year-old Greenwood took his season tally to 15 with a brace against Bournemouth on Saturday.
"The growth that Mason has made in the last year and a half he has been in and around the first team is amazing," Shaw told Sky Sports.
"For me personally, I feel like he has turned from a kid to a man in such short time and now he's understanding what this league is about.
"He has grown in size, he has got stronger and he is not fragile anymore. [Before, he] could be pushed off the ball.
"The most important thing I would say for him is ignore the hype, ignore what people are saying, just focus on yourself, keep doing what he's doing and there is no reason why he can't become a very important player for Manchester United in the future.
"He can become a United legend, no doubt about it."