You're right, a plate is a girl. Specifically a girl that you are dating, having sex with, or intending to have sex with. And also one girl out of several that you are dating at the same time. A plate is one girl out of multiple girls that you are dating. Dating multiple girls at once is called "spinning plates".
It comes from the vaudeville trick where performers would spin plates balanced at the end of poles. The trick was to see how many plates they could keep spinning at the same time, without falling off and crashing to the floor.