Is an idiom the same thing as a hyperbole?

No.

Hyperbole is an exaggeration.

I was so cold last night that my nose turned into an icicle and dropped off.

An idiom is any combination of Words in a language where the meaning is not quite what you might expect.

When Dubya tried to talk French the way he didn't know how to pronounce 'r' stuck out a mile.

'Stuck out a mile' just means 'was very obvious'.