The Solicitor General is often involved in cases before the Court agrees to hear them, by filing briefs either supporting or opposing the grant of certiorari. But in none of the Obama eligibility cases has the Solicitor General filed anything, unless you count a two-line "we don't want to file anything" statement (like the one you posted). Filing something like that doesn't require Kagan's recusal, because it doesn't say anything about the merits of the case. ("We waive our right to respond" is not a legal argument that Obama is eligible, and it's not a legal argument that he isn't eligible.)
Had Kagan filed a brief that says the Court shouldn't grant cert., she would be recused, but she hasn't filed such a brief in any of the eligibility cases.
I'll repeat. None have been heard by the court so none have involved the Solicitor General. What you posted was the filing that will be going to conference and which, if past history is any judge, will get no further.