The internal low-voltage ground is often connected to the 120/220V ground, but it's not always required. (For example, the signal grounds on a stereo amplifier are often connected to "earth ground" through the power cord.)
And also as an example Stereo Hi-Fi equipment in the 70s used only two prong AC cords with no attachment to the household 'ground system'. This had an advantage of allowing the Hi-Fi common ground used by all the interconnect cables between the devices to be isolated from power ground and more immune to AC ground loop caused hum.
So it can be confusing especially for newcomers to electronics. Connecting any two devices to each other requires a 'common reference' connection between them, but this may or not not have any requirement to be household AC ground referenced. AC power grounds primary purpose is safety related to carry fault currents to insure that circuit breakers or fuses open up over allowing 'hot' metal surfaces ready to find a ground path through a unlucky person who happens to be in contact with something else referenced to AC power ground.
Ground is such an overloaded term, often misused or misunderstood to also mean circuit common which may or may not be the case, so almost always requires further information to be fully understood.