You already need to show ID for most medicine! This is such a weird discussion.
It's clearly because you can make methamphetamine out of it. But you can also make methamphetamine out of other things which limiting access to cold medicine demonstrably hasn't prevented.
But the justification is not the same. People were going to the drug store for cough drops and unintentionally ending up addicted to opioids. Nobody was going to the drug store for a decongestant and accidentally making meth out of it, and the number of people getting addicted to pseudoephedrine itself was neither large enough to justify the change, nor its explicitly stated rationale. Nor an effective means to bring it about if the claim that showing an ID isn't a burden is to be believed because it would have no effect on the small minority of people who might go to the pharmacy for a stuffy nose and thereby become dependent on pseudoephedrine, since they would still get it.
Meanwhile the stated rational of limiting availability of methamphetamine hasn't gone well either.
This before we even mention that the OTC replacement, phenylephrine, is not only ineffective, it's a more dangerous drug than pseudoephedrine from a cardiovascular perspective and the switch has plausibly killed some people.
Do you? You have to show an insurance card to file a claim, but I imagine there is a lot of overlap between the people without ID and the people without insurance. And there appear to be free clinics that don't require ID, implying that it isn't required by law.