Foreigners have had to carry "residence cards" (formerly alien registration cards) which are basically these for years.
Having said that, the My Number system is bullshit and will hurt the economy because it will hit people who spend the most money (working cash jobs, side business at home...) disproportionately.
It seems like the foreigner residence cards are only superficially similar. For example, the ID number on the residence card is not a permanent identifier over your lifetime (your number will change when you renew your card). In general, that number is also not directly connected to many other things, not in the way that the government is planning for the My Number system.
Having said that, the My Number system is bullshit and will hurt the economy because it will hit people who spend the most money (working cash jobs, side business at home...) disproportionately.