max current while using a shift register In arduino's official page about ShiftOut https://www.arduino.cc/en/Tutorial/ShiftOut, a drawing with two 74HC595 powered by the arduino board, is displayed. 

Since the resistors are 220Ω, the total current should be 5/220x16 = 0.36A which exceeds the maximum total current of 0.20A for the arduino device.

So, could someone turn on simultaneously 16 LEDs using two shift registers without damaging the device?

Thank you
0.20A is not the maximum current for an Arduino device.  0.20A is the maximum current for the chip that is on the Arduino.  Thus it is only relevant to devices that are directly connected to the Arduino's IO pins (that connect direct to the chip).

The shift registers take their own power from the +5V pin, which is limited to between 500mA and 800mA depending on how you power your board.  Each shift register has its own power limits that are completely separate to the Arduino's main chip's power limits.