Which software is used to produce this video?
My guess is maybe ToonBoom. 

But regardless what software they used to stitch the final cut together, there were a host of software applications involved. 

Illustration, likely Adobe Illustrator and Photoshop, Compositing, Could be ToonBoom, AfterEffects, or others. Audio was done separately, on the tail end. And I jumped over myself, they likely storyboarded all of the animations out in advance, ToonBoom, likely a good choice for that even if it's not what they used for the final mesh. 

In addition, they undoubtably used a host of plugins, perhaps some free, some paid, some very expensive to assist with the motion control and character animation. And last, they likely also relied of pre-existing works, templates, which can be purchased.