I just purchased my 2nd Mavic Pro.
I have one Andriod device (Samsung S7 Edge).
Drone No 1 is set up and flying fine
I use the same android device and application to set up and operate each drone individually but obviously not simultaneously.
I understand that Each controller is specific to each drone however, how does the application behave when there are two different drones.

I am currently getting and error "Check Application" when i power up the second drone.
Nothing more and I do not see what the error is related to.
Is it because I am using the same app for 2 different drones ?

If I create settings for the drone are they captured in the application?
If yes, when I switch drones will the application apply those saved settings to each drone ?

E.g.;
I just finished flying drone number one, I land and powered down.
I exchange controllers and start the second drone.
Will it fly with the same application on the same android device?

I have 2 Mavics .........i think it will have the same settings saved for both drones and not drone specific. At least that's what i am experiencing.

I always shut down the app and swipe it from most recents on my Nexus 6 and Pixel XL when changing them and battery replacement.

I don't get any errors and the RC's i switch them all the time. They are not AC specific at least that is what i experience.

My wife uses a iPhone 6S+ but i haven't used it for both yet.

Settings like MC settings and camera settings will be saved to the drone, not the app, so if you change another drone, the settings you made for the previous drone will not be saved on the second drone even on the same app.
For your case, you can definitely use different drones in one app, if you got "check app" on the LCD screen of the RC only when startup, please wait for a while and check the app until the self-checking finished. If all display well and there is no error message on the app, then no worries.