Let kids be kids. We force so many kids especially girls to grow up so fast it's not healthy. She's 13 not 23. If it makes her happy and it's not harming her then let her be and support her. She probably knows it's kind of weird but so what? She doesn't have to be like everyone else. Check out the ages of American girl dolls they aren't 8 yr old girls a lot are preteen. There's plenty of 13 year old girls out there having sex and acting like 18 yr olds, let her be who she is and have fun. She's creative and has a great imagination and compassion for animals. If it makes you that uncomfortable for whatever reason you have then buy her a real bunny or a dog. Try to channel her energy into what you think might be more productive. Get her to pass out fliers for a dog walking/pet sitting business. Whatever you do stop shaming her and start supporting her.
I made and played endlessly with paper dolls at age 12 and 13. I was interested in being a fashion designer, and got a lot of fun out of the creativity of making outfits for them. If my mother had told me I was too old for paper dolls, I would have been really hurt. Maybe you should back out, or tell your daughter you are sorry about hurting her feelings and she can play all she wants with stuffed animals, and then back out. You do sound too attached to what your child does. She deserves some privacy and some respect for her feelings.
I have 13 year old. Such a funny age. Stuck between little kid and still growing up and maturity comes on at different times for different kids. I personally wouldn't get involved at all in what she does to relax and play in her free time unless it was dangerous. I have a 14 year old that still loves super heros and a 13 year old that builds legos and acts out scenes with them. Fine by me. Better than being girl crazy, on snap chat or as Sandman says, playing video games (which sigh, they do also). I think it was hurtful to tell her she is too old for what she likes. And frankly, want to know the truth? I don't get any but STILL think stuffed animals are awful cute. So, who cares. It's what she likes.
Are you worried about her socially? That would be something we can discuss with you. But she can't help what she likes and shouldn't feel shame for it. I HAVE helped my kids understand that some other kids won't think what they are doing is 'cool' and I'd never want anyone to make fun of them. So, maybe they don't have to talk about how they want to be batman to classmates but at home, it's a safe place to be who you are.
good luck
So would you rather have her playing video games all day long and killing thousands of animals or people in every game? I think that what she is doing in her spare time is healthy. Now granted I only spent 37 years in education with elementary and middle school kids, so am not a real expert on this. But, I think that this probably won't last much longer (which really is too bad, compared to what could happen next). Kids play with legos, rocket ships, army men, and video games. And she still is a kid. That will change.
I think the key is that she wants to be a veterinarian and that is her focus. I am guessing that she has no other siblings and also no pets?
If you want to make a difference, see if you can get her to volunteer to help at a pet store or a veterinarian office and then talk about that.
And finally, you said, "in a deep hole I seem to can’t get out of. " I really am more concerned about you then her. I don't know fear is driving you, but it is something that you need to come to grips with. I hope this helps. Best wishes.