**Core Concept**
The question tests the identification of Neisseria gonorrhoeae, the causative agent of gonorrhea, through its characteristic biochemical and morphological features.

**Why the Correct Answer is Right**
The correct answer is based on the fact that Neisseria gonorrhoeae can be identified by its **oxidase-positive** characteristic. Neisseria gonorrhoeae is a Gram-negative diplococcus that exhibits a positive oxidase test, which distinguishes it from other Neisseria species. The oxidase test detects the presence of cytochrome c oxidase, an enzyme that plays a crucial role in the electron transport chain of bacteria.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because Neisseria gonorrhoeae is not identified by its **gram stain** characteristic. While Neisseria gonorrhoeae is indeed a Gram-negative bacterium, its Gram stain appearance is not a reliable method for identification.
* **Option B:** This option is incorrect because Neisseria gonorrhoeae is not identified by its **methyl red test** characteristic. The methyl red test is used to differentiate between enteric bacteria, and it is not relevant to the identification of Neisseria gonorrhoeae.
* **Option C:** This option is incorrect because Neisseria gonorrhoeae is not identified by its **indole test** characteristic. The indole test is used to detect the presence of tryptophanase, an enzyme that breaks down tryptophan into indole and other products. Neisseria gonorrhoeae does not produce indole.

**Clinical Pearl / High-Yield Fact**
Gonorrhea is a sexually transmitted infection that can cause significant morbidity if left untreated. The Centers for Disease Control and Prevention (CDC) recommend dual therapy with ceftriaxone and azithromycin for the treatment of gonorrhea. It is essential to note that antibiotic resistance patterns can vary by region, making it crucial to consult local guidelines for treatment recommendations.

**Correct Answer: C. oxidase-positive.**