United left-back Shaw scooped both trophies at the club's end-of-season awards ceremony as star team-mate Paul Pogba missed out.
In a difficult season for United – who are set to finish sixth in the Premier League and miss out on Champions League qualification – Shaw stood out among his peers and fans at Old Trafford.
Shaw has made 40 appearances in all competitions heading into Sunday's season finale against Cardiff City – the 23-year-old edging Victor Lindelof and Pogba to the Matt Busby honour, as voted by supporters.
"It's obviously nice for me, but I'd rather win no [individual] trophies and have a better season as a team," Shaw said. "We need to move on and focus now on next season and what’s to come.
"Ole has done an amazing job and of course we've had ups and downs but he's obviously looking forward to getting us back in pre-season and getting his proper work done for next season. We're all looking forward to starting fresh next season with him as the manager."
Shaw, who signed a new contract in October, is the first Englishman since Wayne Rooney in 2010 to win the Sir Matt Busby Player of the Year award.