Factual error: When Jake is given a burner phone by Doug Judy, he receives a text message from the 'Pontiac Bandit'. The phone is an old cellphone from the late 90s, but the text message makes an iPhone sound/tone from the late 2000s. He receives another text message again later with the same tone. (00:09:55 - 00:12:50)
Casecation - S6-E12
Factual error: Amy is wearing a NYPD detectives badge even though she is a Sergeant at this point in the show. (00:01:00 - 00:21:20)
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.