There may be a way to help your issue without scripting, but I'm not promising anything.
First, observe the differences for various file types when you are recording an action Save-As or Save-a-Copy or Export or SFW..., and do a test with and without changing the file's name. Sometimes when you do not change the file's default name or when you do change the file's default name, Illustrator's behavior for overwriting the same file would be different.
If you are successful in getting it to not overwrite one file, it may be possible to achieve your goals by using batch processing and using the 'process all sub-directories' checkbox.
However, if it's not possible with just naming the files different and seeing how the action would play out- you can still get away with minimal scripting - as you would only have to create or find a very simple script to just save your SVG with current file name. Then you can take this super-simple non-batch script and include it as a menu item in your batch action! At that point, you should be able to do the exact same action process (process all sub-directories) and it should work exactly how you want!
Here's a sample script for you to try.
function test () {
var doc = app.activeDocument;
var newDocName = doc.name.replace(/\..+$/, "") + ".svg";
var svgOpts = new ExportOptionsSVG;
doc.exportFile(File(decodeURI(doc.fullName.toString()).replace(doc.name, newDocName)), ExportType.SVG, svgOpts);
};
test();