Why do none of the 5 have real names?

Because...laziness? Honestly I think if they gave the characters proper Chinese names then every voice actor would have to learn how to pronounce it properly and have consistency across all of them, which I think would be unlikely. I once watched the dubbed version of an anime and only one of the actors pronounced the main character's name correctly. It was straight up painful hearing the others mangle the name.

Also, consider the audience: children. Imagine unique, non-descriptive names for all the other characters. I could see it causing a lot of confusion. So they took the easy route: easy main character (Po), descriptive names for the Furious Five, and two syllable names for three others, and you're all set. It also meant they didn't have to put in effort coming up with more Chinese or Chinese-sounding names.

They are each named after a Kung Fu fighting style.

Tigress uses Tiger Claw Kung Fu System
Crane uses Fujian White Crane Kung Fu
Monkey uses Monkey Kung Fu
Viper uses Snake Kung Fu
Mantis uses Northern Praying Mantis Kung Fu;

"I'm the dude, playing a dude that's disguised as another dude".

They're not characters, they're kung-fu style stereotypes personified.

In any case, think about 'Sifu' - I think it should be pinyined as 'shifu', but other than that, it's like calling your doctor or master 'sensei', it's not a real name, either.