Variables - Data Set name from list

No, it's not possible. But as an alternative, I would rename my data set names, since the names can be used to set the file name at export time.

renaming data sets can be a pain if you have lots of them or if you do it often. 

this script renames data sets in a jiffy

// rename Data Sets as the FIRST variable data value
// https://community.adobe.com/t5/illustrator-discussions/variables-data-set-name-from-list/td-p/15480644

var idoc = app.activeDocument;

var vars = idoc.variables;
var dsets = idoc.dataSets;

for (a=0; a<dsets.length; a++) {

    idoc.activeDataSet = dsets[a];
    
    dsets[a].name = vars[0].pageItems[0].contents;
}

@maria_3014 Honestly, I don't think Illustrator native Data Sets panel have a built-in function to use a variable as the output file name when you batch-save or export. The default output names are either "Data Set 1", "Data Set 2", etc., or they are based on the original file name with a sequential number.