The new Weyland identity Becasue We Build It says:

1 [Recurring Credit] Use this credit to advance ice.

And the encounter clause of Matrix Analyzer says:

When the Runner encounters Matrix Analyzer , you may pay 1 [Credit] to place 1 advancement token on a card that can be advanced.

Can I use the recurring credit on the identity to pay for the Matrix Analyzer clause if the target is an ICE ? I want to say yes, but the phrasing on Matrix Analyzer is a little weird.

I'd say no. You're using Matrix Analyzer's ability, not advancing ice. (The end result is that ice ends up with an advancement token on it, but it's not quite the same thing.)

Hypothetically: What if Matrix Analyzer said "you may pay 1 credit to place an advancement token, draw 1 card, and initiate an end the run trace." You get to advance ice, but here I think it's more clear that what you're doing is paying for a conditional ability that is distinct from the "advance" action.