Generalized data structure
Notice that a data structure stores more than just it's accessible elements. This is probably what mean by additional feature, but I would consider it the essential feature. Take for example a binary search tree. This tree stores information on the order of its elements. In particular, by a traversal you can output the sorted sequence of elements stored in the tree in linear time. There are roughly $\Theta(4^n)$ different full binary trees with $n$ leaves. Thus the  data structure stores roughly $2n$ additional bits in this case.

Maybe this view helps. Some algorithmic problems are one-shot problems. So you take the input, process it, and then (hopefully) output the solution. Other problems, however, are trying to solve several problems for which part of the input is mostly the same. Here it might pay off to preprocess the static input (the data), such that further requests are easier to answer. The result of the preprocessing is stored (often in form of a labeled graph). This is what you do, when you store data in a data structure.

The above paragraph explained how static data structure behave. Usually data structures are dynamic, allowing you to alter the represented data and its associated information.