genital organs are part of the reproductive system.
the other word in plural doesn't mean several times but several examinations acorrding to differents criteria in one time check. (IMHO)

The man's writing is not accurate in form or in content. Formally, there is no such thing as الفحوصات. There is فحص (singular), and there is فحوص (plural).

Content-wise, the man most probably meant "tests" rather than "examinations." He was probably asked to undergo several tests, the nature of which he obscured by the euphemistic expression للجهاز التناسلي, which he probably used out of embarrassment or out of desire to write (or speak) in a respectful manner. The result is that the specific nature of the tests is now unclear.

Military academies typically require tests that rule out STDs (sexually transmitted diseases). In some cases, where the schools are strictly for male students, especially in countries where military recruits are male only, students must prove that they are males. This legal requirement would necessitate a medical statement (how the statement is actually substantiated may vary from place to place).

Which word should you choose in this context will depend on the nature of the tests, which you can verify through your client. In the absence of definitive information, a literal translation of the words as stated will suffice.

The reproductive system is called the "genital system" (as it pertains to the function of "generation"), and the "sexual system" as well. In males, it is also called the genitourinary system because of the combined functions of some parts.

Masculine Reproductive Organ
Since the question is about a man. The term must be masculine specific, not generalized.