I think both fists can be seen in other Chinese martial arts as well. The phoenix eye seems like an important tool in Southern Praying Mantis (although it'd probably be best if a practitioner could confirm this for us) and the ginger is present in Choy Lee fut. Not sure if this is true for all the families though.
As for how it's trained, I'd say through a lot of form/technique training, drills, and conditioning work. Similar to how we'd train energies like finger jabs or any technique that would cause too much damage if done the optimal way (attacking soft tissue, vital points) but not enough damage if just applied to any part of the body (ex: like finger jabbing a person in a knee).
That's my take on it, although I'm sure there are many differences in opinion as I've met others who actually even have different ways of forming fists such as the phoenix eye. The variations were mostly based around the placement of the thumb. For example, the way I train, the thumb is kept bent about 90 degrees at the middle joint (if that makes sense, sounds weird explaining it like that) behind the index finger while some people I've met had it so that the thumb was bent so that the tip of the thumb was tucked into the "pocket" formed on top of the index finger as a result of the way that it is held in the phoenix eye. This gives them a second striking point that is formed by the middle joint in the thumb sticking up (although both methods have more than 1 striking point from what I've learned). Now that i've typed this out i realize that it probably sounds strange. so i just took pictures of my own hand, sorry if it looks strange, never really took pictures of my fist in such a manner before lol.
How I train my phoenix eye:
How some friends have trained their phoenix eye:
Some may bring up things about which fist is more likely to break under impact, but personally and from my own experience I'd say that it's all dependent on the way one trains and uses the technique, and that can vary a lot.