From the link provided in this comment it seems that the broker is in the US, and indeed is in Illinois. As the answer by user6726 correctly says, there is no legal or regulatory requirement in the US denying individual Syrian citizens brokerage or other financial services. Any such denial is the broker's own policy, and is not mandated by law.
A person is probably required to truthfully answer all questions on an application for such services. But if the application only asks for a single citizenship, it is probably not requires to list all citizenships an applicant may hold.
Moreover, the Illinois Human Rights Act (775 ILCS 5) may well apply. 775 ILCS 5/1-102 (A) declares the following to be the public policy of the state:
Freedom from Unlawful Discrimination. To secure for all individuals within Illinois the freedom from discrimination against any individual because of his or her race, color, religion, sex, national origin, ancestry, age, order of protection status, marital status, physical or mental disability, military status, sexual orientation, pregnancy, or unfavorable discharge from military service in connection with employment, real estate transactions, access to financial credit, and the availability of public accommodations
Section 775 ILCS 5/5-101 (A) defines "Place of Public Accommodation".
(A) Place of Public Accommodation. "Place of public accommodation" includes, but is not limited to:
...
(6) a laundromat, dry-cleaner, bank, barber shop, beauty shop, travel service, shoe repair service, funeral parlor, gas station, office of an accountant or lawyer, pharmacy, insurance office, professional office of a health care provider, hospital, or other service establishment;
Whether a broker's office is an example of "other service establishment" is not absolutely clear, but the tendency has been to interpret this to include any facility which generally offers its services to the public.
Section 775 ILCS 5/5-102 provides that:
It is a civil rights violation for any person on the basis of unlawful discrimination to:
(A) Enjoyment of Facilities, Goods, and Services. Deny or refuse to another the full and equal enjoyment of the facilities, goods, and services of any public place of accommodation
{Boldface in all ILCS citations added for this answer.}
Thus it might be unlawful discrimination to deny services based on "national origin" in Illinois. There is a similar US federal statute, but its definition of "place of public accommodation" is significantly narrower, and it does not appear to apply to circumstances as described in the question. Note that "citizenship" is not specifically listed as an unlawful ground of discrimination. But "national origin" is defined as meaning the country in which a person or that person's ancestors were born. To pursue such an anti-discrimination claim, one would probably have to apply, be refuses, and sue the broker. Quite a bit of trouble and expense. A lawyer would be needed to have a reasonable hope of success.
If an application is made listing "Mexico" as the applicant's citizenship, that may or may not be a "material omission" when the applicant actually holds dual citizenship. It might be wise to consult a lawyer with knowledge of securities law on this point.