While Eobard is in his cell talking with Barry, he recounts his origin story, which is taken from the 1990's version of Thawne's comic book origins, ignoring both his original origin and his rebooted Rebirth origin. Thawne started out obsessed with The Flash and wanted to become like him, replicating the accident that gave Barry his powers. After traveling back in time to meet his hero, Thawne discovered that he was destined to become Flash's greatest enemy, and became resolved to be superior to Flash in every way, thus becoming the Reverse-Flash.
Cisco says to Reverse-Flash, "Bye Felicia".
It is from a scene from Angela Means' character Felicia [spelled as Felisha] in the American comedy film Friday (1995). It is 'to get anyone out of your face'. It is generally intended as a dismissive send-off.
At the end, Patty rings Barry and says about a man in the train with a gun. The Flash reaches the train. There was just no time in between that Barry could have contacted the Flash, had they been different persons, explained the situation to the Flash, and then the Flash could have reached the train. This way, Patty confirmed for herself that Barry himself is the Flash, and understanding that Barry is not accepting just to keep her safe, she still goes away from Barry's life.
Patty was a CCPD detective so she had access to enough resources and clout that she could have kept a trace on Barry's mobile number and thus, could have known of all the hundreds of instances when Barry and the Flash happen to be at the same location, confirming that they have to be the same person. Of course, it would have involved the ethical aspect on keeping a trace on a friends mobile number for non-official purposes.