Is it permissible for a girl to take donation money given to help her with her studies after she concealed her identity, and would she consider it a debt upon her if it is not permissible?

The safest course of action for your friend is not to take the money for herself. She should return it to the donors immediately or give it to someone who fits the description of the donation. If she strongly believes that the donors would not have donated had they known the truth, then it is not permissible for her to take it. Her lie to one of the donors about the actual academic year makes it unlawful for her to take their donation if that donation was specifically intended for that year. However, if the academic year was not the donor's primary intent, then the lie does not necessarily prohibit her from taking it. She must repent to Allah for intentionally lying.