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I mean what the reason we are able to use those methods?

Still Unclear

equals() and toString() are overrides by String Class, and they belong to Object class

You're confusing final with private.

toString(), equals() these methods are from Object class( super class of all classes)
So any class will get these methods because of Inheritance

We don't need to extend a class to call it's methods. We just need an object of that class.

For example:
String obj= new String("abc");//This creates an object of String class which we can than use to call methods in String class.

Hence final has no event in this regard.