My son did not like the texture of meat either when he was littler. He ate mostly chicken nuggets, roasted chicken in chunks, and occationally hot dogs and fish sticks. He would not touch any type of beef or pork. He's outgrown most of it, he is five. I think that the texture kind of puts them off. It could be worse, my niece would not eat anything but sweets, chicken nuggets, milk, corn and french fries between the ages of 3 and 7.
I would not worry about having the same meat item with your child. We keep a package of purdue chicken nuggets in the fridge and he ate those, with whatever veggie or other sides that I fixed for the family meal. Or if she likes them, you could keep a few boiled eggs in the fridge to have instead, and if she doesn't like the meat, she could have an egg. It was not that much extra effort. In time hopefully she will try more. When she gets older, it will be easier to talk her into it.
That said, small children do not need as much protein as we probably feed them. If she drinks a lot of milk she's probably getting most of her daily requirement from that. One good brand of milk is farmland dairies, it has about 11 grams of protein per serving, which is about as much as a child size piece of chicken. Other good sources are cheese, beans and tofu. They even make meatless chicken nuggets that are usually available where you find the Tofu. They are absolutely delicious. My son loved them when he was smaller. I would think in terms of total grams of protein a day. Get in touch with your pediatrian, or just search the internet for "daily nutitional requirements" and the USDA recommended daily requirements should be easy to find. This will put your mind at ease.
I'm not here to convince you that you should be vegetarians, because obviously you are not, but if you are concerned that she is not eating right, perhaps you can get a good kids veggie cookbook that will teach you how to make balanced meals for your daughter, that the whole family could enjoy on occation, or a least as a side dish that could be enjoyed by the family and eaten as an entree when the family is eating something that she does not enjoy.