You are correct. The corrected sentence should be
She is the same girl who complained about the food in the mess.
Using "that" instead of "who" in this context is an extremely common error in English. So common that very few people will notice it, either in written or spoken English. However, questions testing whether you know this show up frequently on standardized tests like the SAT and GMAT. A very educated person will always use "who" in this context, certainly when writing.
Don't ask me which one I use most often or you won't respect my education. ;)
BTW, best not to use "a/c" for "according." It either means "air conditioning" or "alternating current." Outside of these contexts it will only cause confusion.