removeElementAt() method is available in vector class so,its also available to stack class ..but using this method any element can be eliminated..which cannot be a proper implementation of stack... or is it?
No, it's not ideal. Those are classes from the original version of Java, and if they were added now I suspect they'd be designed differently - I'd expect to see Stack as an interface, with some class (possibly one of the existing collections) implementing it. A bit like the way LinkedList implements the Queue interface nowadays.
Agreed. This is one example of incorrect inheritance. A stack is not a vector, so Stack should not extend Vector. Similarly, a property list is not a hash table, so Properties should not extend Hashtable. In both cases, composition would have been preferable. There is one item also listed in Effective Java related to this.