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It is only possible to sync one single catalog. So it won't work assuming a different catalog on the second PC.
It may be simpler to log-in to your LR Mobile Webspace and download photos within a browser.
For the future you can of course copy a catalog backup to the second PC and copy all the images if you have a clone e.g. exactly the same folder & file structure.
Is there a way to download files in dng or xml of those files?
It is only possible to download the smart previews either individually or as a batch in a zip folder. To export as DNG you need access to the masters. You would need to re-create the catalog on the second PC and update the folder locations where necessary.
I guess it's the only way for me as right now there are proxy copies of files in Lightroom web.