How does an Android app accesses any browser's history?
That app uses Accessibility service (originally meant to assist users with disabilities) to read what is being displayed on the window. It also uses app usage permission to track what app is being used at a time. Combine the two features and the app can know when a web browsing app is active, and what is being typed in there and displayed there. That app has already declared in its description on Play Store that it uses the said service. Under Web Content Filtering &amp; Web Monitoring heading, it states:


  This app uses Accessibility service to limit app usage and keep your children from internet dangers


Related:


How do certain apps record everything we type in android?
Unusual ways of using Android accessibility services (by Maksym Dybarskyi)