In nearly every western country (EU countries, Canada, Australia, New Zealand etc) the samples are securely stored indefinitely - in the US it depends on the state but many do keep the samples there as well. That does not mean they are readily accessible- there have been cases of misuse but it is extremely rare.

From a DNA perspective one can only use the Guthrie test samples for about a decade after collection so yes, in a worst case abuse scenario we have cards that could be mined for DNA across a bunch of <10 year olds.