When browsing around in Chrome for Android, Lastpass pops up with suggestions if it recognizes the URL as one you have associated login details with.
How does it kno
I general using javascript, apart from Andoid functionality, and with some hacking, it is possible to get references to all open windows (and for instance get their URL). See this answer from 2009.
LastPass on Android asks for Accessibility permission with canRetrieveWindowContent
true. This lets it traverse the current view hierarchy, and access the views as AccessibilityNodeInfo objects.
Accessibility API lets you search nodes by the text displayed on the view, and also gives you java class name of each such view. AccessibilityNodeInfo#findAccessibilityNodeInfosByText
The feature doesn't work on Opera probably because the devs never handled the view hierarchy traversing logic for it.
Check out https://developer.android.com/guide/topics/ui/accessibility/services.html