Read the URL of currently open tab in a browser, every time it changes
Apps on Android can't access data of other apps. Therefore you can't just read the current URL of a web browser like chrome. 

If this would be possible every app with advertisements would do so and report all the URLs you are visiting in the web browser back to some ad network. As you can see such an interface would be a privacy nightmare.  

Hence the only chance to get what you want to do it inside of a browser-plugin.
However I don't think that there is any browser-plugin for Chrome, Firefox or another web browser on Android that would do what you want. Hence you would have to develop your own web browser plugin (please note that development topics are off-topic on this site).