When Spider-Man 2099 first arrives in Earth-67, the police car behind him reads N.Y.P.D. 5087, before the camera cuts to a close-up of the 67 Spider-Man. When it cuts back to 2099 and for the rest of the scene, the police car only says NYPD.
When Peter Parker, Miles Morales and Gwen Stacy are standing outside Aunt May's door, Peter starts walking away until Gwen fires up a web to pull him back. In the very next shot, Peter is back in place but the web behind him is gone entirely.
In the fight between the Spider-Men and the villains at Aunt May's House, Aunt May knocks Tombstone outside with a bat. Seconds later he can be seen briefly in the background fighting in the house again.
In Doc Ock's office, the camera cuts to Miles after Spiderman asks what he's doing. It reveals he's carrying both the monitor and computer tower. When Olivia Octavius says she can't wait to watch Parker suffer in pain, the camera cuts to Miles again, but he's only carrying the tower, not the monitor.
Miles' dad lives in Brooklyn, where he gives Miles a ride to school, yet when the villains show up at aunt May's house, which is in Queens, he picks it up on the radio and drives over. However, it's never said that he is a Brooklyn cop - nothing prevents him from living in Brooklyn and driving a police car there, while working in Queens, especially since PDNY probably serves all the boroughs like NYPD does.
You famously can't unlock the back door of a police car from the inside, yet Miles gets out of his father's squad car unassisted. It is plausible that this is just one of many difference between our Universes. It does make certain sense to give officers the ability to control whether the door can be opened from inside or not, just like in most modern private cars.
Miles blames himself for 'being followed' to Aunt May's house, when by then it was public knowledge that May's nephew was Spider-Man (perhaps even earlier, if the brightly glowing, emblem-showing Spider-Cave entry door in the back yard gave it away). However, the villains also knew that they were dealing with entirely different Spider-People and total strangers to aunt May (at least in this dimension), so they wouldn't see a reason to search for these new Spider-People at May's place.
Miles has Gwen's hair stuck to his hand after they shave her head to free them. In the next shot when he's walking through the hall, his hands are clean. As we know, the "sticky hands" effect comes and goes quite suddenly, so it's plausible that once he left the office, the hair got unstuck at some point.
During the final battle Miles takes away Wilson's gun, stating that it would be cheating. If a gun is cheating, then web slinging devices should be too (both are tools used against foes). However, one crucial difference is that a gun is a lethal weapon, while a web slinger is not. Moreover, Miles was obviously joking when he said that (a typical Spider-Man behavior in battle) and didn't seriously intend to introduce any "rule of engagement", if only because evil villains wouldn't accept any such rules and Miles is smart enough to understand that.