The PushOwl widget is for making sure you don't lose out on potential subscribers. It displays a message asking your store visitors, who might have previously blocked your notifications, to subscribe again.
But Chrome, Firefox, and Safari do not offer support for receiving notifications in Incognito / Private modes. This is probably to keep the privacy issues in check.
Following is an image showcasing the difference between the permissions of an Incognito / Private mode (left) and that of a regular browsing mode (right).
Since store visitors using Incognito / Private modes cannot become subscribers, the PushOwl widget does not serve any purpose for them. This is why the PushOwl widget is not displayed in Incognito / Private browsing mode.