Why can a 74HC595 drive 8 LEDs?
You can (and should) run the LEDs at lower than the maximum current. Many modern LEDs are plenty bright at a few mA, no need to run them at 20mA unless you are planning to cast shadows across the room or blind unsuspecting users.  

Exceeding the absolute maximum Vcc or GND current is not guaranteed to kill the part immediately, but it will probably affect reliability negatively. You could always use a 74LVC595A (100mA abs max) but it's still bad practice to get close to the absolute maximum values.