Honestly not trying to be flippant, but safe is a very relative thing. Your portable power inverter must come with a grounded 3-prong, 120VAC charging input.

Unless it's old and not updated, your house likely has 3-prong grounded outlets throughout. The ground pin is wired back to your panel where the neutral wire is also connected. Both neutral and ground are bonded together and then tied to the planet using a water pipe and/or a grounding rod. Again, 2 of the 3 pins in your home outlet are physically connected together right in your panel. So if they're connected together, why bother having a redundant wire? Because two wires (hot and neutral) are reasonably safe, but 3 wires (hot, neutral, and ground) are SAFER. If anything bad happens, you have an EXTRA conductor that is normally not used and is designed to protect you from shock. Often grounded appliances have exposed metal that can be touched while the appliance is in use. The ground pin likely connects directly to the exposed metal so that if something inside the appliance goes horribly wrong and the chassis becomes energized, the grounded chassis will protect the user from being shocked.

So as you discovered, your grounded 3-pin device still works fine using just 2 pins. Is it as safe as having that 3rd pin tied to Earth? No. But as you discovered, that 3rd pin isn't used for anything unless something goes horribly wrong. There might not even be any exposed metal on the outside of your portable inverter. If that's the case, then there shouldn't be any way for you to be shocked through casual contact with the chassis of the inverter. But your inverter also has its own outlets on it. The manufacturer has no idea what you'll be plugging in. So for safety, they likely connect the input ground to the output ground so that when your inverter is plugged in to your house outlet, any grounded device that you plug into the inverter is also grounded.

So by carrying the earth ground from the charger input, through the inverter, and to it's outputs, it has maintained a ground connection throughout.

Carrying the ground connection everywhere is one line of thought, but there is another.

Another slightly less obvious solution is to do the opposite. The opposite meaning to specifically avoid ANY grounding. Everyone knows that for current to flow, you have to complete a circuit. Birds can land on high voltage power lines because a circuit has not been completed. Their bodies are instantly elevated to thousands to volts compared to ground when they land on that exposed wire, but they don't explode in a puff of pink feathers because they have not completed a circuit and so no current is flowing and they're "safe". Similarly, utility workers can be helicoptered onto live wires to perform maintenance while the line is energized for the same reason. My point is that if you can isolate the power from the ground, then that's another way to help avoid getting shocked. Power plants are grounded, so there's no choice there. But what if you could produce your own power and NOT reference it against ground? If there's no connection between your power inverter and the Earth, then standing on the ground and touching the live output wire will not complete a circuit. You'll still get shocked if you connect yourself between the hot and neutral pins (you've completed a circuit), but many electrical shocks are between hot and ground. So if the inverter does not have any connection to ground, then touching hot while standing on the ground does not complete a circuit.

Again, "safe" is a very relative thing. Your inverter was designed to be grounded while charging and plugging it into the Ridgeline inverter breaks that connection. I have a pure sine wave inverter that I use all the time while camping. It has 3-prong outlets and has the neutral and ground pins connected together. But it has a 12V DC input without any connection to ground. So that ground pin is a bit fictitious because any 3-prong appliance I plug into my inverter is absolutely not grounded. But at least ground and neutral are connected. So if something goes horribly wrong inside of an appliance I'm using and the hot wire comes in contact with the chassis, then at least that chassis is wired back to the inverter neutral and the inverter should go into overload and shut down. In the case of the Ridgeline, there is no ground pin. So if something goes horribly wrong inside of an appliance and the hot wire comes in contact with the chassis, the chassis will become electrically live compared to neutral, but because there's no ground reference in the system, the whole assembly is not live compared to Earth. What I don't know is if the Ridgeline has any electrical connection between the inverter output and the body. I doubt that it does (my guess is the inverter output is isolated), but I do not know.

It's a bit confusing, I know. I'm sure you wanted a simple answer, but I can't provide you with one because it's more complicated than that. Plus, my experience is that by far most people do not understand the intricacies and will give poor advice. When handling power, there is ALWAYS some level of risk. If you understand the risks and how any given wiring scheme exposes you to those risks, then you can decide for yourself what you're comfortable with.

 

It seems to work well but wondering if this safe to do?

Short answer: No.

I suggest charging your power station from one of the 12 VDC power outlets instead. I don't expect the charging time will be much different than using AC charging with the Ridgeline's outlet in 150 W mode. The quality of power produced by the Ridgeline's outlet is AWFUL. I wouldn't use it for anything I cared about.

Typically, grounded power cords are used for equipment that is not double-insulated.

Not using the ground conductor increases the risk of electric shock.