Looking at the first supplied video, it looks like they are using some form of IK and that is driving the majority of the animations. Unfortunately Godot does not provide any IK support currently, though it is possible to write your own IK setup using GDScript.
In the second video it looks like they are using animation blending and IK stuff for placing the feet. It looks like what they did is make a animation for moving forward and moving sideways and then blended them together based on which direction the player is moving in.
Personally I would suggest making two animations, one for walking forward and another for walking sideways, and then blending them as needed. I personally don’t know how to blend animations in Godot, but it is possible if this tweet is any indicator. For moving from forwards to backwards you could make a rotation animation and blend that with the forward movement animation, or just rotate the player as the walking animation plays.
Using an IK system is also possible, but may be much harder as there is currently no IK support, so you’d have to write your own (I’m have my own IK system and it only sorta works)