Is the theory of 3d finite element method just an assembly of 2d finite element analysis by putting planes on top of each other, or, a much more comple and different theory applies for 3d, with respect to 2d? For example I am reading a dam design manual from 1970s, where computers were low power and FEM was gaining new acceptance. It says that 3d analysis can be done by making 2d and then stacking planes on top of each other. But now, after 40 years, is it still the same logic for 3d, or now 3d has its own theory itself, by utilizing much better computer power?
In solid mechanics, the essential difference between 3D and 2D FEM is that 3D FEM approximates ("discretizes") the 3D elasticity equations and 2D FEM is based on the elasticity equations simplified to two dimensions. So understanding the applicability of 2D or 3D FEM really is about understanding the assumptions of the two elasticity theories.
I have never seen structural FEM explained in terms of "putting planes on top of each other."
I suggest you take a look at this site: solidmechanics.org. It begins with a thorough presentation of the elasticity equations, includes an introduction to the finite element method, and includes some sample codes for 2D and 3D FEM.
The theory of 2D and 3D finite elements is quite similar. In a 2D setting you will chose a Triangulation of your domain (tetrahedral/hexahedral grid), then introduce proper base functions defined on those elements. In 2D FEM, you typically couple your cells degrees of freedom with those neighbouring it. So if you have a rectangular 2D grid with one dof per cell, then that might couple with four other cells producing a line in your system matrix with five entries. In 3D your coupling increases, as your cell has more direct neighbours.
When you say "Stacking 2D planes onto each other", you have to be careful to respect the coupling which will be in the vertical direction.
It might be instructional to start with 1D FEM. Do that discretization "by hand" and write down the resulting matrix. There are a couple of examples out there which have five elements or so and it can be done in approx. 30 min.
From there you may move on to higher dimensions. The differences and the similarities will become clear.