Why does the electron diffusion current point towards the N side when electrons are diffusing to the P side from the N side?
In non-technical English, a current is the flow of anything, and it's direction is the direction in which the substance in question moves.
In the language of electronics, there is something called conventional current. In the case of holes, the direction of conventional current is in the direction of the movement of the holes. In the case of electrons, conventional current is in the opposite direction to the movement of the electrons.
Thus, holes and electrons, each moving in opposite directions, create the same direction of conventional current.
The moral is: don't think of conventional current in electronics as the same as the non-technical meaning of the English word current.
Also a follow up, in a non-uniformly doped semiconductor, shouldn't the diffusion current point towards the direction of the region that has a relatively greater concentration of charge carriers.
[As an aside, on this site, you generally should not expand your question once it has been answered.]
Holes will diffuse from an area of higher concentration to an area of lower concentration. Since holes are positively charged, the conventional diffusion current composed of holes will be from the region of higher hole concentration towards the region of lower hole concentration.
Electrons also diffuse from regions of higher concentration to regions of lower concentration. But because electrons are negatively charged, the conventional diffusion current composed of electrons is in the opposite direction from the motion of those electrons. So the conventional diffusion current composed of electrons flows from the region of lower electron density to the region of higher electron density.
Remember this is backward from usage of the term "flow" in nontechnical English. But in electronics, we use conventional current, which is in the opposite direction from the movement of electrons.
since the diffusion, without the influence of any electric field, is driven by the repulsive forces between similarly charged ions, so shouldn't the region with more concentration of charged ions diffuse to the other side more, and hence contribute to the total diffusion current more than the other side?
Diffusion current is driven by a gradient in carrier concentration, NOT by the repulsive forces of similar charges. Neutrally charged particles also diffuse, like molecules in a liquid or gas. However, since they are neutral, there is no electric current associated with such diffusion.