The data live on your eID, which is basically a Yubikey except you can’t save whatever you want on it.

Original page, since I don’t want to paste the full translation: https://meid.minv.sk/?page=frequent-questions%2F#howItWorks

Some countries in the EU like Slovakia also allow you to use passkeys that basically act as supplement to using physical ID card, so that you can use something better like Yubikeys or save it to your password manager.