If I have two Skinshifters in play, can I use the ability of one to effect 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: