This sounds like a lazy state legislature that doesn't much care for the 9th and 10th in this regard.
If their law really did merely rubber-stamp "whatever federal alien-registration laws" might be in effect, then the State would be committing itself to enforce (instead of ICE, DHS) whatever immigration registration Congress hands down.
If that is the intent, then as others have posted, AZ law enforcement could've been directed to do same and then instead sued the feds to pay for it. No state legislation needed to bind their hands.
The correct path would have been to file a law suit in federal court against ICE and the Obama Administration for failing to enforce immigration laws with due diligence. I suspect that would have had more legs in the SCOTUS than this attempt to usurp federal powers.
Another way to approach it is to increase penalties for driving without a valid license and for document fraud, and other crimes and infractions that are popular with illegals.
Public transportation is a bigger problem but could be improved by selling passes at a discount to those who show a valid ID in person, making those without a valid ID pay more. Or better yet, make a valid ID a requirement to use public transporation for safety reasons.
Trying to enforce Federal immigration law without the Feds cooperation can cause difficulties.