TL;DR Hard to show someone lied if they told a variant of the truth
Race is not black and white - it is many shades of gray. There are a lot of scenarios where I may call myself, by one definition, "Asian", and you may call me "white". Or vice versa. In OP's example, someone might have facial features and hair that are commonly characteristic of "Asian" and a typical "Asian" last name. Yet that same person might have been born outside of Asia and therefore arguably not be Asian. Not a "lie". Just a different definition of the truth.
To put it another way: If the defendant goes into court, I would suggest that rather than say they lied, they instead say "I don't call myself a real Asian." Then under cross-examination it will come out how the plaintiff is defining "Asian" - e.g., at least 50% Asian ancestry (making that up, but plausible) - and the defendant would then be asked if they fit that definition: "Yes, my parents were born in Asia." "Then you are Asian! You lied!" "No, I was not born in Asia. I consider myself an American. The sign didn't say 'Asian ancestry', it said 'Asian', and the way I understand 'Asian', I am not one."
In the US at least, "race" is an extremely controversial term, for a lot of reasons. It is also an extremely vague term. Consider how many different places we are asked, typically on government forms, political and other surveys (yes, that includes StackExchange), applications for all kinds things. Sometimes that is for marketing reasons, sometimes for pseudo-legal reasons (to show that a particular "thing" is non-discriminatory by showing a (self-identified) distribution across various races/ethnicities/eages/etc., sometimes for no apparent reason at all except "all the other surveys ask it". But as it is commonly self-identified, consider the possibilities. I'll use "Asian" as an example. Which of the following are included:
- Born in Asia
- Born outside of Asia from two "Asian" parents
- Born outside of Asia from one "Asian" parent and one non-Asian parent
- Born outside of Asia but of Asian ancestry and grew up in Asia
- Born outside of Asia but of Asian ancestry and grew up in an Asian community - e.g., Chinatown in many US cities.
- Born outside of Asia but of Asian ancestry and grew up outside Asian in a non-Asian community (and if so, when does Asian ancestry cease to be a factor)
Arguably in the example case, unless someone was born in Asia they could argue, if it came to a legal battle, that they do not consider themselves "Asian" and therefore were not misrepresenting themselves when signing the contract.
In addition, what is Asia? I think the term as stated in this question likely refers to China, Japan, India and nearby areas. But according to Wikipedia it includes the Arabian peninsula (Israel, Saudia Arabia, Iraq, Iran, etc.), Russia (not clear how much of Russia - but definitely a lot of it) and many other places that may not fit the common "Asian" reference.
As another example, "African American" is a common term used in similar situations. There are other terms. But using "African American", if a "white" person from Egypt or South Africa immigrates to the US, are they "African American"? Clearly not the intent of the categorization (which is based on very real both historical and ongoing discrimination against specific groups of people) yet linguistically and biologically correct.