Oh she is SO adorable! You are doing amazing caring for her.
She is definitely a goose! She looks a lot like my girl did at that age (an Embden mix), and we also only had her for the first 4 months of her life, no other geese. There are definitely people out there who make one goose work, but we found it pretty much impossible. I admit, I did dote on her, maybe a bit too much but it broke my heart to hear her cry when we tried to leave her by herself. Maybe if she got used to it younger, it would have been okay. I wish that we got her a friend when she was younger, because she truly has no interest in other geese... she tolerates him at best, though having him here does still make her feel more secure. I have also seen single geese who are happy because they have a kitty or dog friend (be extremely careful with intros - even the sweetest old dog may attack a tiny gosling), do you have any other pets?
In your pictures, she looks very young, maybe 2 weeks old. She doesn't look 3 weeks old yet, that's when they start looking a little awkward!

I will attach a couple pictures of my girl at 2 weeks old, so you can compare. She may grow up thinking that she is a human, like ours did. There is a big difference between her and our second goose who spent the first couple weeks of his life with other geese.
Like others said, they unfortunately do not have the muscle control other animals do to control where they go. Diapers are not bad in and of themselves, and they are pretty important for a goose spending time indoors. I spoke to my vet about my initial concerns with diapers (who I trust completely as he is extremely knowledgeable about geese - a rarity) and he assured me they are perfectly ok to use - but to make sure that they fit well, that they are changed often, and that the poop isn't left stuck to her feathers. She will probably need her bottom rinsed a few times a day. It's best to get a diaper meant for a goose, not one meant for chickens. I make our diapers myself, because my girl is very finicky and we tried all the ones available online and she was never perfectly comfortable in them and I felt that they could be improved. (Though our other goose never had a problem with most of them.) If you have access to a sewing machine, I would be happy to walk you through making them, but it's probably best to try one made from a company like The Goose Mother first. Goslings go so often, and also have much wetter/runnier poops, so they need their diapers changed very often. When she is older, it should be a lot easier to deal with.
Another thing we used was PUL fabric, but you will still have to wipe the poop off that. It will save a lot of money vs. puppy pads, though. The fabric is meant for making (human) baby diapers, so it is waterproof. You can drape it over your lap or get big enough sheets to go on your bed, etc!
Okay, that was longer than intended - sorry! I hope this has helped a bit! Please feel free to message me if I can help in any way, I know just how tricky it can be raising a single gosling, indoors too, and there isn't always a lot of answers on to be found through googling. Thanks so much for adopting this sweet little creature. She is clearly in great hands!