Using PEX 1.23.4
Spigot 1.8.8

When I do pex user aplayernamehere group add anyothergroup and then check the player by using pex user aplayernamehere it shows the added group only.

Then I used pex user aplayernamehere group set default to restore the original situation.
Next I used pex user aplayernamehere group add twice to add this player to two other groups. Then the added groups as well as the original group showing up correctly.

I checked the permissions.yml file by hand and the default group was not shown on the users groups.

This is expected behavior in 1.x -- the default group is not actually stored as a group for the user.

So basically group add has the same effect as group set when trying to add a 2nd group?

It has the same effect if the player only has the default groups, because of how the default group is handled in 1.x added at runtime if the user has no groups in the permissions file).

I am not sure if there is a misunderstanding but maybe default was a bad example for the default player group. This could be any group name.
So the player was assigned to a group - lets call it newbie this time - when I tried to add the user to a 2nd and 3rd group.