The first video is a cut-out animation. This technique can be achieved in Blender by rigging textured planes.
Regarding the setting, those planes are distributed in 3D space to achieve depth. This is enhanced by the use of depth of field through a perspective camera and other effects.
A recent example is Nikolai Mamashev's rig inspired in the comic Pepper&Carrot by David Revoy:
This covers at least the character animation. Effects such as smoke, particles, and image post-processing can also be composited in Blender.
The second video might be a vector animation. Here the assets are drawn in a vectorial software. Even though the technique is similar to cut-out, it also allows to keyframe the vectors for a smooth animation such as the one for the hair.
As far as I know, this could not be achieved in Blender as easily as the previous example. If you are looking for a Free Software package for the task, I recommend Synfig.