How is this circuit a voltage regulator?
The circuit is a voltage regulator because the output voltage \$V_o\$ does not depend on the load. Whether you connect a \$1~\Omega\$, \$1~\text{k}\Omega\$, \$10~\text{k}\Omega\$, etc., resistor from the output to GND, the voltage at \$V_o\$ will remain at the voltage set by the expression you've derived. The current \$I_o\$, however, will be different in all these cases. In other words, it regulates the voltage \$V_o\$ based on the values of \$R_1\$, \$R_2\$ and \$V_Z\$.

The "opposite" circuit is a current regulator, which keeps the current \$I_o\$ at some value regardless of the load connected to it (and the voltage \$V_o\$ changes accordingly to make this happen).