Software used to create such animations?
You could recreate most of the stuff in that video inside Adobe After Effects, if you have all the assets on individual pictures (layers), and not merged on top of each other.

You can easily take a layer (image file) and independently change the position, rotation, opacity, size, etc. with keyframes. If you have the body parts of a character as different files, you can position them how you'd like them by dragging and dropping, and then parenting them to one specific layer, for example, the chest. Then when you change the position of the chest, the entire body moves with it.

It would also be a good idea to have the transparent versions of the flat icons, so you can add a custom background, and no whitespace overlaps with other layers.

It would be a better idea to draw any characters inside AE with the shape tools so you can modify the shape of any of the body parts.