If I have two Skinshifters in play, can I use the ability of one to affect the other? For instance, if one of my Skinshifters has Ice Cage on it, can I use the other Skinshifter's ability to morph the "caged" one, thereby breaking the Ice Cage?
No, you can't.
First and most straightforward, the Skinshifter's ability does not require a target, and Ice Cage requires the enchanted creature to be targeted by a spell or ability for it to break.
Second, you can use Skinshifter's ability only to change itself, not another Skinshifter.
If a card says "Something happens to CARD NAME", then this always references the card this text is printed on, unless explicitly stated otherwise. If a card references another card by name, the syntax is always something like "A card/creature named XY", for example like Llanowar Sentinel, or Angel's Herald.
The rules for that is this:
201.4. Text that refers to the object it's on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.