Settlement agreements are contracts like any other. Generally speaking, their promises are only enforceable by and against the parties to the contract. However, courts will sometimes allow non-parties to enforce a contract's provisions, if they conclude that the nonparty is a "third-party beneficiary."
It appears the Giuffre-Epstein settlement was entered into in Florida, so Florida law would govern the question of whether Prince Andrew was a third-party beneficiary. To settle that question, Florida asks whether the third party is merely an incidental beneficiary or the agreement, or is instead "a member of the limited class which was intended to benefit from the contract.” Technicable Video Sys. v. Americable, 479 So. 2d 810, 812 (Fla. Dist. Ct. App. 1985).
Here, the parties' public filings don't include the exact language from the settlement, but they seem to suggest that in exchange for some payment from Epstien, Giuffre agreed in 2009 that she would release "all claims against him and numerous other individuals and entities," and that that group included unnamed "royalty." But in 2020, Giuffre entered into a revised settlement with Epstein's estate that appears to have modified those terms.
Your question is just the one the court is currently considering. Can the court permit the case to go forward despite the existence of this agreement? The answer may depend in large part on whether Prince Andrew is a third-party beneficiary:
Prince Andrew argues that he is a third-party beneficiary of that agreement because that was what Epstein intended and because releasing him would have benefitted Epstein by protecting him from entanglements in further litigation.
But Giuffre argues that Prince Andrew cannot be a third-party beneficiary because neither party intended to release him, because he couldn't have been a defendant in the the Florida case, and because the 2020 agreement abandoned the 2009 settlement and specifically preserved the claims against Prince Andrew.
Because the court has (inappropriately) permitted the parties to keep the agreement under seal and redact relevant arguments from their briefs, it's hard to make any predictions about which way the court should rule. The primary question will be what the parties intended when they entered into the agreement, but the most important piece of evidence of their intent -- the agreement itself -- is not yet a public record. That leaves us unable to reach any conclusions about how the court should rule or whether its rulings are consistent with the law.
As a general contract-drafting principle, though, well-advised clients who intend to provide for third-party beneficiaries will typically include language explicitly acknowleding those beneficiaries, the parties' intent to confer a benefit on them, and the nature of that benefit. It doesn't sound like that kind of language exists in the 2009 release, so I'd expect Prince Andrew to have an uphill battle with this argument.