As mentioned by others, the reason why get(), etc. is not generic because the key of the entry you are retrieving does not have to be the same type as the object that you pass in to get(); the specification of the method only requires that they be equal. This follows from how the equals() method takes in an Object as parameter, not just the same type as the object.
Although it may be commonly true that many classes have equals() defined so that its objects can only be equal to objects of its own class, there are many places in Java where this is not the case. For example, the specification for List.equals() says that two List objects are equal if they are both Lists and have the same contents, even if they are different implementations of List. So coming back to the example in this question, according to the specification of the method is possible to have a Map<ArrayList, Something> and for me to call get() with a LinkedList as argument, and it should retrieve the key which is a list with the same contents. This would not be possible if get() were generic and restricted its argument type.