No, it can only handle one phone at a time. This is an understandable shortcoming that I think all car phone kits or radios have.
What I didn't expect was that you have to tell it anytime you want to use a different phone.
My Parrot in my other car works the way I would like for the AVIC to have worked. You can pair up to four or five phones with it (only one at a time), and you set one of those as primary. Every time you start the car, it searches for the primary phone. It uses it if it finds it. If it doesn't find it, then it used the next [previously paired] phone that it does find. With this setup, it works well and automatically when I drive the car with my phone, or when my wife drives with her phone. (If we are both in the car, then it always picks mine.)
Of course, that still wouldn't do what the OP wants to do.