OMG U R SOOOOO RIGHT
I don't believe that the book means the Order of the Phoenix. We knew Flamel was a friend of Dumbledore, as well as an ally of the Ministry. I couldn't tell for sure, but the witch in the book appeared to be Seraphina Picquery, so it appears he was an ally of MACUSA as well. It is likely similar to the Order, but not the official one. As the Director said in CoG, Albus had a lot of allies, and Flamel was one of them. These allies are what eventually became the Order of the Phoenix in the wake of the First Wizarding War, but they weren't considered an order back then. They didn't meet and discuss things regularly like a real entity. It was a coalition of allies with emergency ways to contact one another, but that's all.