Alarm clocks aren't working on Android 6
When you say the little alarm status bar symbol, do you mean up the top left? If so, it sounds like that alarm app is running as a Service rather than registering with the system's AlarmManager.

When you swipe the app away from the multitasking view, you are killing the Service. More details

I have only a few suggestions:


Ensure the app cannot be optimised to save battery. Settings->Battery->Battery Optimisation (Overflow menu). Ensure app is displayed in Not optimised section.
Don't swipe the app to close it. This typically isn't needed on modern Android phones. Though really, I'd still not trust the alarm as the system can clean up apps whenever it needs to.
Use a different alarm app. I use Timely on Marshmallow and have never had a problem with it.