carry algebraic data like the Zariski topology on a ring (spectrum)? If they exist, what are they and how are they used?

In model theory they define and study the 'algebraic data like the Zariski topology' irrespectively of where these data come from. These data are called Zariski geometries, and e.g. admit some intersection theory, can be used to prove Chow's lemma, admit some classification results in dim 1 etc. You may want to have a look on the recent book of Zariski geometries and references therein (or the actual book Zariski Geometries : Geometry from the Logician's Point of View, by Boris Zilber).

Also, the book has some examples which sometimes need some work.