Courts of law established by the federal government must comply with Article III of the US Constitution (state courts, which is where the overwhelming majority of all controversies in the US are heard, don't need to be structured along the lines of Article III, because the US Constitution defines the federal government and not the state governments).
An Article III court is staffed with a federal judge appointed by the President (as specified in Article II) with a lifetime tenure. Article III courts "extend to all cases, in law and equity, arising under this Constitution".
FISC court judges are appointed from the federal bench (they're judges appointed by a President for a lifetime term on a federal court). But they have a limited tenure on FISC --- seven years. And unlike Article III appointments, FISC judges are appointed by the Chief Justice of the Supreme Court. And, most importantly, FISC judges do not have power that extends to all cases in law and equity under the Constitution. In fact, the only thing they're empowered to do is authorizing FISA certifications.
Things that the FISC does not appear to be able to do: hear criminal cases, sentence defendants, overturn laws, or establish precedents binding on any other court.
There are other federal courts that aren't Article III courts --- bankruptcy is one of them. Bankruptcy court judges have 14 year terms. Even though they function as part of the district court system, the Judiciary is clear that they aren't Article III judges. It's worth noting that Bankruptcy's bizarro status in the court system has also created stare decisis problems with the rest of the court system. There's little reason to believe that FISA courts, which are even further removed from Article III than Bankruptcy, would be more judicially potent.
One dispute about the Constitutional status of FISC judges arises from case law in the 1990s, wherein a defendant claimed that the FISC was a violation of separation of powers. The appellate decision ruled that FISC judges weren't violating Article III by serving on the FISC, but I dispute that the language in the decision dictates that those judges are in fact Article III judges; I think it says merely that a federal judge doesn't violate Article III by sitting on a body that is effectively an agent of Congress. If they are Article III courts, the FISC is unconstitutional (in a technical way, not a substantive way).
I'm a layperson summarizing independent research here, so I'd very much appreciate an expert's opinion on this analysis.