Is she the one in you avatar? SHES SO CUTE!! I love GSD, they're one of my top breeds for a future dog, once I'm living somewhere I don't have to worry about breed bans/renting. And she has one of my favorite colorings- I love the darker ones as opposed to the ones with a smaller saddle/mask. And thats a great age to get a new dog, you're through the puppy crazies/adolescence but you still get the tail end of the puppy wigglies, as I like to call them. And I love the name!
I also have depression and really bad anxiety and I got a new pup in November- it was the best decision I ever made. She gets me out of bed in the morning and gets me on walks most days (ie when its not pouring or really cold, she's a Boston Terrier so she's not super weather tolerant unfortunately, another reason I like GSD, they're a dog anything kind of dog). She may very well sense it- some dogs are known to be natural alerters and can sense panic attacks coming on in their handlers. Do be mindful that with a protective breed like this, though, that your panic can "travel down the leash". Part of anxiety is us seeing threats where there are none, and this is a breed that has been bred (at least recently) to be mindful of the handler and react to perceived threats.
An "emotional support animal/dog" is really no different from any other dog. This is opposed to a "service dog" which is a highly trained dog proficient in high level obedience skills (ie, able to maintain an hour long down/stay at a restaurant without anyone knowing its there) which is also task trained to assist a disabled person (this may be in many different areas, examples are dogs taught to pick up dropped items, wake from nightmares/panic attacks, taught to brace their owner for mobility on command, or seeing eye dogs, among others). Pretty much, "emotional support dog" just means dog.
"emotional support animals" are granted housing protection (ie, allowed in public housing and 'reasonable accommodations' must be made by landlords in private housing, although that's a definition open to interpretation under the law) but their owner must be legally disabled. They aren't granted public access like service dogs, and they don't have to be specially trained. Recently a lot of people have been trying to pass their dogs as "emotional support animals" and bring them into public, which is illegal, or are not disabled and still call them that, which is also illegal.
My guess would be her old owner mentioned something about it and the shelter wrote it hoping she'd be adopted. It's possible she has some link to a service dog program and "washed out" for not being 100% perfect (they wash dogs out even for very minor things because service dogs have to be 100% reliable in public), but I would guess the place would have tried to place her themselves rather then send her to another shelter. Sorry I ended up writing so much about that, I'd recently looked into the difference myself and learned a lot.