Hello, everyone! I’m new here.
This is Dakota. She is an AKC Registered (full-blooded) golden retriever that we got from a breeder. We got her as a puppy and expected her to grow out long fur, but never did. She is 2, soon to be 3 years old, but still hasn’t grown out a long coat. Why is she short-haired like this? Is it possible the vet made a mistake and registering her as an AKC golden retriever?
Dakota is a doll, I especially love her color

If you ever shop for a purebred dog again in the future, it's important to understand that the ONLY thing that AKC registration means is that the dogs who are the puppy's ancestors are registered to as that breed with the kennel club. The AKC does not check to see if they are actually good representatives of the breed, if they have good temperaments or if they follow the physical 'standard' agreed on by the breed club members. There is no requirement to breed AKC registered puppies other than the parents being AKC registered. If Dakota's breeder isn't trying to breed puppies with longer fur, than chances are the puppies won't have that type of fur. They may have had different priorities than fur type or they may have had zero priorities and just wanted to breed their dog.
Here is the link to the breed club standard for Goldens:
AKC Breed Standard - Golden Retriever Club of America if you read the first paragraph, you'll notice that the standard says Goldens are "primarily a hunting dog" and the personality should be "eager, alert and self-confident." Those are the big picture things that you need to have a good Golden. Here is what is says about coat:
Coat — Dense and water repellent with good undercoat. Outer coat firm and resilient, neither coarse nor silky, lying close to body; may be straight or wavy. Untrimmed natural ruff; moderate feathering on back of forelegs and on under-body; heavier feathering on front of neck, back of thighs and underside of tail.
Coat on head, paws and front of legs is short and even.
Excessive length, open coats and limp, soft coats are very undesirable. Feet may be trimmed and stray hairs neatened, but the natural appearance of coat or outline should not be altered by cutting or clipping.
Form follows function and your girl has more of a correct coat for performing the job of a hunting dog than if she had a ton of long, fluffy fur. She should have enough undercoat to protect her from cold water, but not so much fur that she can't shake and dry off.
I think Dakota is adorable and I'd rather have her amount of coat than the type that is flowing almost to the ground. In the future, if you have a certain idea in your head about what your breed of dog should look like or act like, be very sure the breeder you choose to buy from is trying hard to produce those kinds of dogs. Especially in Goldens there are, unfortunately, a big split in the breed's priorities and some focus more on the outside of the dog and some focus more on the inside of the dog. Moderation across the board is key. When you see dogs you like, ask people who the breeder is, especially on this forum, you may find a breeder you'd like to go with in the future.