Some Android phones/versions provide the ability to personalize the look & feel of the application icons (square, circle, teardrop, etc.). This system setting is usually found by long-pressing on the Home screen then selecting a "Home Settings" or "Personalize" popup option etc.
Is there as an API that we can use to programmatically determine what that IconShape setting is currently set to?
This system setting is usually found by long-pressing on the Home screen then selecting a "Home Settings" or "Personalize" popup option etc.
That is not a system setting. That is a feature of the launcher app that you happen to be using. There are hundreds upon hundreds of launcher apps, both ones that are pre-installed by manufacturers and ones that are installed by users. Not all launcher apps have this feature.
Is there as an API that we can use to programmatically determine what that IconShape setting is currently set to?
There is no Android framework API for controlling this. You would need to contact the developers of the launcher apps of interest to you, asking them if they offer an API for this for their specific app.