TLDR:

• Minecarts can derail if at least 2 are on same track, bump each other and possibly accelerate (needs more testing)
• Reproducible for Minecraft 1.13 snapshots (noticed in 17w50a), confirmed for 1.12.2 by Xavom and panda4994
• Derailing happens in the curves of a track
• Derailing (of 1 of 2 carts, 1 containing an entity incl. player) seems to only happen with rails laid out in East-West direction, unless you have 3 carts on a track in North-South direction; derailing positively confirmed in this case for 2 carts containing mobs, 1 cart empty-
  Currently unsure about directionality, in 1.15.2 pre-1, derailing happens for me at the moment solely in North-South-direction, if 2 entities are in a minecart, one can derail. (needs more case testing)
• Derailing does not seem to happen if it's 2 carts only (with no entities) on a track as it seems you can't make them accelerate/not bump each other enough , they rather seem to drive together like train waggons (needs more testing)
• Empty carts which are on neighbouring rail lines, laid out East-West show a different behaviour than empty carts on neighbouring rail lines, laid out South-North; it seems the minecart collision box differs here, see video:

Also empty (not containing an entity) Minecarts can derail if there are at least 3 of them on a track;
this includes also the North-South-direction.
Currently suspected this is the case because in this setup they may not all be driving in the same direction, but one of them will be driving in the opposite direction, and the upfollowing "bumpings" or suspected accelerations may propel one of the carts out in a curve.
The same suspected accelerations seem to be happening if one of the carts stops and gets bumped by one of the others.

Because of MC-92165 I host I've got various different cart test tracks spread on my several bug test worlds; I noticed a derailed cart in 17w50a and after conducting some simple tests, this can be reproduced relatively fast. Use mc-123367.nbt in 17w50a or mc-123367_1-12-2.nbt in 1.12.2 to test it yourselves.

The derailing of 2 carts on same track only, one containing an entity (incl. player), seems to be directional (but needs more testing to be sure).
 
Short video proving this behaviour with one test conducted, see above.
Longer video with several tests in the same manner can be watched here: https://youtu.be/NdC3-KfTXls

I also noticed that when carts bump into each other, they seem to be more prone to simply stop their movement if rails are laid out in East-West-direction. This happens usually at the curves of a track (watch also Youtube video to see it in action).

I currently didn't see any derailing when it was 2 minecarts only, with no entities inside one of them, but more tests have to be conducted to verify this. This may be because of a needed gain in acceleration of a cart for derailing, which seems to happen by "bumping" into each other, which seems to be more often the case if one of them holds an entity (maybe due to the entity's hitbox sometimes bumping the other cart into the other direction).

Note that in the test track the 2 rail lines got no gap between them, which could maybe cause an interference between the minecart's and some entities' hitboxes.

Derailing happens in the curves of a track, same like the carts sometimes stopping their movement after bumping against each other.

Currently, I don't seem to be able to reproduce the derailing (pig in 1 cart, 1 cart empty) if the tracks are laid out in North-South-direction.
The stopped movement of the cart also seems to occur much more rarely with North-South rail tracks.
Needs more testing.

Additional info by Unarybit:
If you build rails like in the picture with a mob in a minecart on it and the player stands inside the "rail ring" at the height of the mob collision box, he is pushed around. It does not happen with just the minecart or outside the "rail ring". The effect is more visible when you fly.

The player is pushed around when the minecart is in the N / E (white) corner and the player is in the area of the white blocks.
The player is pushed around when the minecart is in the N / W (black) corner, and the player is in the area of the black blocks.
As you can see in this video: