It has sweet FA to do with "free food"

Yes it was. The bands were not given to all immigrants, only those with an entitlement.
Some form of identification is necessary.

IT WAS COMPULSORY TO WEAR THEM AT ALL TIMES, INSIDE AND OUT - IT WAS A FORM OF TAGGING - NOTHING MORE

That is the point of wristbands.
They stay on.
At hotels they cut them off when you check out.
No-one else can use them because cutting invalidates them.

They can not be lost or stolen, unlike cards.