using UnityEngine;
 
public class ClipBoard
{
    public static void ExportString(string exportData)
    {
#if UNITY_ANDROID
        AndroidJavaClass jc = new AndroidJavaClass("com.unity3d.player.UnityPlayer");
        AndroidJavaObject activity = jc.GetStatic<AndroidJavaObject>("currentActivity");
        activity.Call("runOnUiThread", new AndroidJavaRunnable(() =>
        {
            AndroidJavaObject clipboardManager = activity.Call<AndroidJavaObject>("getSystemService", "clipboard");
            AndroidJavaClass clipDataClass = new AndroidJavaClass("android.content.ClipData");
            AndroidJavaObject clipData = clipDataClass.CallStatic<AndroidJavaObject>("newPlainText", "clip text", exportData);
            clipboardManager.Call("setPrimaryClip", clipData);
        }));
#endif
    }
}

環境によってはcom.unity3d.player.UnityPlayer をcom.unity3d.player.UnityNativePlayerなどと変更する必要があるかと思われます。