Hello, valterra,
Hey, you've reached the point where you start to wonder about all the things everybody else seems to take for granted. Electrickery is totally mysterious, but "It just works".
If you hold up your small inverter, that device is designed to be a "mobile" and "floating ground" electrical device. What that means is that it is not intended to be a permanent part of a building. It is only meant to be installed in a vehicle, or temporarily where it will not interface with other electrical power supplies.
The wiring in your house, supplied by the grid, is designed to be permanent, grounded, and protected from a variety of fault conditions that the mobile inverter cannot do. It achieves this by grounding the enclosures of all receptacles and fixtures with a "non-current-carrying" ground conductor (the bare wire). The electrical circuit is carried entirely within the two insulated conductors. All black wires (or red if 240V) will have 120VAC supplied to them, and they must all be circuit protected at the box. All of the white wires must lead back to the box, and when no load is present, there must be no voltage applied to them. In the box, all of the white wires are "tied" to ground. Single point grounding of the white return wires ensures that they all have basically zero voltage. Those bare ground wires still have to be there, to protect against faults, lightning, water (or other contaminants) causing shorts, even though the do not play an active part in the normal electrical circuit.
Back to your inverter: it likely has a grounded case. From its point of view, "ground" is half-way between the black and white wires. This allows a certain amount of safety, but it does not permit grounding either the white or black wires. The white will have voltage, like the black, so you cannot short it without releasing the magic smoke.
Could you wire up a shed, in isolation, with that inverter? Perhaps, but it is not as safe. Twice as many wires have power, so are you going to put circuit breaker on both wires? Where is the circuit protection in the inverter? It is built into just one breaker. If you wire up the shed with Romex in several separate circuits (one set of lights here, some receptacles there) then what protects each circuit? It's the same nerve-center breaker that wipes out everything. If it's a fairly high capacity inverter, say 1800W, then it will take about 20 Amps to flip that thing. That 14 gauge romex isn't rated for 20 Amps.
It goes on and on... I'm only scratching the surface.
5 years ago, I didn't know any of the stuff I just wrote, either. This hobby has turned into an education for me, too.
Have a nice weekend.