Because she has spent a thousand years hiding inside Eientei, Kaguya is often depicted in fanworks as a recluse or even a hikikomori who shuns outside interaction (which is not entirely accurate). In humorous works, she is often portrayed as being a NEET, or unemployed geek, who sits at home and surfs the Internet all day.
In contrast to above, her portrayal in Inaba of the Moon and Inaba of the Earth shows her as a sheltered person who wants to explore the world outside of Eientei.
The NEET portrayal is also depicted having a crush on Cyborg from DC by a well-known fan of both Touhou and DC; Touhou Myotanpen plays with this because Cyborg is an actual character in that fanwork.
Popular nicknames for Kaguya among her fans in Japan are "Teruyo," which is a deliberate mispronunciation of the kanji in her name, and 「ぐや」 ("guya") which is an unusual abbreviation of her first name, ommiting "ka."
Kaguya's appearance closely resembles that of Hazuki, the cat-ear-wearing vampire girl from the anime Moon Phase. Because of this, Kaguya is often depicted with cat ears, either real or fake, on her head, even occasionally uttering lines from the anime's opening song, "Neko Mimi Mode". The cat-and-mouse dynamic between her and Mokou adds another layer to this depiction.
Since she is generally acknowledged to be easier than Remilia, Yuyuko and sometimes even her servant Eirin, Kaguya is thought to be rather weak for a final boss (although most bosses in Imperishable Night are considered easier than those in earlier games, and continues are not allowed against Kaguya, unlike the previous two). As a result, fanworks often depict her as being completely helpless and dependent on Eirin to survive, and her theme song's climactic passage has had the words "Eirin, Eirin, tasukete Eirin!" (Eirin, Eirin, help me Eirin!) set to it by the fans. However, Kaguya has a larger spellcard arsenal than either Remilia or Yuyuko and attacks confidently and without hesitation in the actual game. In addition, considering she didn't make the Touhou Hisoutensoku roster for being overpowered, Kaguya being weaker than other bosses might not necessarily be true. Indeed, if one takes the Final A route in Imperishable Night, it is actually Kaguya who aids Eirin for Eirin's final spell card.
She is commonly referred to as simply "princess" by several fans.