So now 911 is getting two calls, tying up two operators, who have no way to know that they are the same incident. So two emergency response teams will be dispatched.
No, it doesn't work that way.
Phone rings. 911 operator #1 starts putting information into their CAD (computer aided dispatch) system, straight-away, including victim information. 911 operator #2 sees that there is an incident already in progress at that location, communicates with operator #1, and adds their own information from their own caller.
In the event that #1 and #2 submit overlapping information independently and simultaneously, they'll soon be merged...often by a supervisor.
Meanwhile, in your view, every time someone calls 911, another squad gets dispatched. It'd be comical if it worked that way, so it's probably a good thing that it doesn't.
The reality is that when someone falls ill/there is a car crash/a fire/whatever-folks-call-911-for, several calls are usually made to 911 -- sometimes, dozens. Fortunately for you and me and that other guy somewhere else in the city who is dying and needs an EMT, they're very good at sorting this stuff out.
It's what they do for a living.