Hey Nathan Duncan
It appears as the app you're working with is not specifying their launch mode to open as a single task. This may be an intentional choice by the application you're integrating with. More details below...
After looking at closer, it appears like the application you're working with from Explorer is launching an activity that has standard launch mode | Android Developers This allows the activity to be opened within another apps context. This is useful for some activities, such as the Camera, that can be used inside other apps without having to actually switch to the camera app. Now, depending on the application your opening this may be intended, but based on what you're describing it sounds like there is some undesirable behavior.
I can easily reproduce this by creating a sample app what has an intent filter set up for "https://myapplication.app" for an activity that has a standard launchMode. If I switch the launchMode to "singleTask" (first line of code below), the activity is launched within its own task (not Explorer).
<activity android:name=".MainActivity" android:launchMode="singleTask">
<intent-filter>
<action android:name="android.intent.action.MAIN" />
<category android:name="android.intent.category.LAUNCHER" />
</intent-filter>
<intent-filter android:autoVerify="true">
<action android:name="android.intent.action.VIEW" />
<category android:name="android.intent.category.DEFAULT" />
<category android:name="android.intent.category.BROWSABLE" />
<data android:scheme="https" android:host="myapplication.app" />
</intent-filter>
</activity>
Joel