Yes it’s illegal, but …
First, we will deal with the easy bit: if you knowingly employ deception (stating something on the application that you don’t believe is true) in order to receive an advantage (admission to college), then that is most definitely fraud.
However, action against the applicant by the college or police is problematic and probably won’t be pursued.
Race in US law
Race is certainly a legal construct in US law but the statutes that use it, don’t define it. It is therefore for the courts to define it in the context of the particular case according to current societal understanding.
This is a pretty thorough overview.
The National Academies of Sciences, Engineering & Medicine define race as “a complex concept and viewed as a subjective social construct based on observed or ascribed characteristics that have acquired socially significant meaning.”In other words, race is socially defined by grouping people into distinct groups based on real or stereotyped physical features (e.g., skin color, hair texture).
Most likely for college admission is this self-referential definition.
The U.S. Equal Employment Opportunity Commission (EEOC) define race and color as they relate to discrimination:
Race discrimination involves treating someone (an applicant or employee) unfavorably because he/she is of a certain race or because of personal characteristics associated with race (such as hair texture, skin color, or certain facial features). Color discrimination involves treating someone unfavorably because of skin color complexion.
So, given that there is no objective “test” for race but merely a moving social consensus, it would be necessary for the college to prove that the persons self-identification of their race was a deliberate lie. That is, they, personally, don’t actually believe they belong to the race they self-identified.
While objective evidence that they had, before and since, not identified as that race almost certainly exists it would be difficult to find and not definitive as a person’s self impression of their race may change over time. This change could be the result of external stimuli (e.g. a DNA test or learning more about their ancestry) or it could be purely internal: they felt they were that but now they feel more like this.
Even if it was a bald faced lie (a statement of fact without belief), if the person persisted in that lie, including perjuring themselves, there is no real way to prove the lie. However, a judge or jury may still decide that they don’t believe you (whether or not you believe yourself) and find there was fraud.