Shaw's debut campaign at Old Trafford has been interrupted by persistent injury problems, making 13 Premier League appearances since joining from Southampton for a reported £28 million.
"It's been a frustrating one because of the injuries I've picked up but I’m loving it [at United]," he told Match of the Day magazine.
"If I was to write a school report on my first season here I'd give myself a C-.
"You don't realise just how big Man United is until you're actually here – it’s frightening. It is the biggest club around.
"I’m playing with the likes of Wayne Rooney, [Angel] Di Maria, [Radamel] Falcao and it's weird because my friends always talk about them, saying how good they are and I'm just thinking "Wow, I play with them".
"I'm probably one of the luckiest 19-year-olds to be alive."
United visit leaders Chelsea on Saturday, but Shaw is a doubt with a hamstring problem.
