She is widely known as "China" (中国, pronounced Chuugoku), the one whose real name is never remembered.
In fanworks, she doesn't take that well which leads her being shunned or defensive of her name.
It is known that in 2ch, there was a debate on how to read Meiling's name. One side said Hong Meiling, and the other said Kurenai Misuzu. The debate ended (sort of) when a user said "Why don't we just call her 'China', as her clothing seems to identify her the most." As a result, even after ZUN said "Hong Meiling" is the correct pronunciation in an interview, people kept on calling Meiling "China".
Meiling's nickname is further ingrained into the minds of fans by her clothing, and also by the name of her stage BGM, "Shanghai Teahouse ~ Chinese Tea". Her stage theme tends to be more frequently remixed by fans than her boss theme, "Shanghai Alice of Meiji 17", to the point where "Chinese Tea" became her theme in Touhou Hisoutensoku.
A popular Flash animation called "My Name is Hong Meiling" spawned the meme "Sakuya-san, Sakuya-san, cleaning stop! Room cleaning stop!" (after which she gets stabbed in the forehead with a knife).
She is commonly referred to as the "lazy guardian" of the Scarlet Devil Mansion, though whether this stems from her actual personality or from the nickname "China" is unknown. Many Japanese people complain about the reliability and quality of products manufactured in China. There are many fanfics that depict an enraged Sakuya behind a sleeping Meiling in front of the gates to the Mansion. The implication is that she has the "Chinese" quality of "Shinrai no Chuugoku" (信頼の中国, which translates to Reliable China), with much intended sarcasm.
Her relationship with Sakuya seems to be a popular topic for fan work, including comics, music and games. One running gag involves her being stabbed in the head by Sakuya's knives. This is usually due to her inability to keep unwanted visitors, mostly Marisa, out of the mansion. In canon, Meiling doesn't appear to be chastised for failing to keep Marisa out (canonically, she is known to be rather versatile at her job and the witch is the only known person whom she can't stop from intruding).
It is also common for fan works to depict Meiling caring for Sakuya when the latter was a child. This scenario is somewhat plausible due to Meiling being a youkai and Sakuya being human, making it likely that Meiling is several decades older than Sakuya.
Additionally, they both have the distinction of being the only characters in Embodiment of Scarlet Devil to have braided sideburns with ribbons. This fact supports that setting.
In the MUGEN community, Meiling is known for her "Chuugoku Satsu" attack, an obvious parody of both her nickname "Chuugoku" (China), and the signature attack "Shun Goku Satsu" (Instant Hell Murder) of Akuma from Street Fighter. There is also a version of her called Lie Meiling, which seem to be an alternate reality version of her, that uses much less colored attacks.
As the result of her comedic scenario in Touhou Hisoutensoku and the nature of one of her battles (with a rather simplified background image), fans have made numerous jokes regarding drug experimentation on Meiling's part.
In fanworks, Meiling seems to be some sort of big sister or playmate to Flandre (rather than Remilia). In comedy works, it can be to throw Meiling in the way of more danger. In serious ones, her laid back approach to her job and uncontrollably friendly nature make her a natural choice for reaching out to an otherwise isolated Flandre. Many fan works depict the two going out of their way to help each other on numerous occasions.
Her race is Youkai, but she's commonly associated with dragons due to her many shared motifs with them, such as her use of rainbows, hat, and name, and possible inspiration by the Chinese Hong. This is most prevalent during depictions of her theoretical EX mode in which she often adopts dragon characteristics.
In some fan works, Meiling is known to say "JAOOOO" (じゃおおおお), a war-cry attributed to her. This cry originated from the 2nd Touhou M-1 Grand Prix (Comiket 73).