I believe you are referring to the Parallax Scrolling technique.
Parallax scrolling is a technique in computer graphics and web design, where background images move by the camera slower than foreground images, creating an illusion of depth in a 2D scene and adding to the immersion.
Referring back to the screenshots in OP, this appears to be the technique in play. In order to create the 3D depth illusion a number of images are layered, with the bottom most images on top. Each layer will scroll at a different speed.
To simply the scenario, consider the game screen is split into 3 images:
- The background
- The ground
- The foreground
The foreground is layered on top of the ground, and the ground is layered on top of the background. The background will scroll slower than the ground. The ground will scroll slower than the foreground.
Consider the screenshot in the bottom right.
The grass is in the foreground, and will move across the screen much faster than the dirt pathway as you walk left/right. Also, it will appear over the road and any characters in this plane. The mountains will scroll much slower in comparison to the dirt pathway. Additionally, anything in the plane of the dirt pathway such as characters will appear over the background.
I have used an example of 3 planes, but you will probably find the background has multiple planes itself (e.g. the clouds may move slower than the mountains). The main ground will also have its own plane system so that items lower down the axis will overlap items higher up. To complicate matters slightly, this is different from a character jumping (they will move higher up the screen but remain in the same axis/plane).
As mentioned in the comments, this technique is part of the 2.5D technique.
The Parallax Scrolling technique is used in a large number of games. Streets of Rage and Street Fighter spring to mind. It is also implemented into the SCUMM game engine (although this game engine additionally offers sprite scaling on the y-axis).