Yes, 120 LEDs * 60 mA = 7200 mA = 7.2A
An Arduino Nano will use 19 mA (Arduino Nano Tech Spec). Of course, if you add a power LED, or other components it will add, but it will likely be less than a few 100 mA. So that totals to max 7.4A.
It's always good to have some headroom which is 0.6A in your case (that's enough).
Actually what's happening is as follows:
- Your 8 mA gives the current the components request
- Your Nano requests 19 mA
- Your LEDs require 7.200 mA.
As long as the voltage is the same (5V for both the LEDs, Nano, and adapter), and the components do not request more than 8 A summed up (with some headroom), the adapter will deliver the current needed. If all components request more than 8 A, than high likely the adapter will break (and possibly the attached components too).