Does an ability that refers to the name of the ability’s card have to refer to that instance of the card?
The untap Soliton isn't a targeted effect; it only untaps the Soliton which used it. When a card's name is used it really means [THIS-CARD]. If it said U:Untap target creature named Soliton, then you could untap any Soliton, but it just says Untap Soliton, which translates as Untap ~this card~.

You will still clone the ability though, and one will resolve before the other allowing you to untap your Soliton, then tap it to mill 2 cards, clone the mill ability to mill 2 more cards, then untap it again, and mill 4 more cards! So you get 8 cards for 1 mana instead of 2 cards for 1 mana.

If you start with a tapped Soliton, it works like this:


U:Untap Soliton added to stack
Bracer's Fires, adding a second untap Soliton to the stack (Has to be the same card)
Let one untap resolve, stack still has 1 untap pending
tap soliton to add mill 2 cards to the stack
a second mill 2 cards added to the stack
4 cards are milled, then pending untap fires, soliton is untaped
tap soliton again, add mill 2 cards to stack
bracers add a second mill 2 cards
4 cards are then milled, stack empty.