Written in 2023
Cat people have been around japanese culture for quite some time. They are supposedly based on bakeneko yokai, divine creatures that look like cats or have feline features.
Later on in the 19th century, the folklore spread and there were rumors and talks about bakeneko prostitutes in Japan.
Cats are considered cute, elegant and independent, and they are among the favorie pets around the whole world. Being cute and admirable (or rather: being admired for being cute) in japanese culture is called "moe" and has created some sort of moe culture, and many cat girls in popular animes and mangas of the 80s and 90s were depicted as cool and admirable, and so they gained some popularity.
It's not solely a japanese thing that catgirls are prominently displyed as being admirable. The famous "Cat Woman" from the Batman series or Tigra from Marvel. In fact, Tigra appeared as early as 1972, although she missed some of the cat girl features that are so prominent in Japanese role playing games these days like actual features from cats within her physique.
In the anime and manga culture, the trend of displaying cat girls has significantly cooled down, but in games, the trend is still going strong.
There is the famous Miqo'te race in Final Fantasy XIV, which is very popular among players, and many run around as cat girls (but also as cat boys). They are humanoid with cat tails and ears. There is the Woren tribe from Breath of Fire, who also feature the famous cat ears and tails, but the male woren look a lot more like anthropomorphic animals than just humans with some cat features, while the females are more reminiscent of typical catgirls, although their lower body is also covered in fur.
Genshin Impact features the Kätzlein (which could be translated to "little cat" in German, although the word per se doesn't exist in everyday language) bloodline, Xenoblade Chronicles the Gormotti race - it's in so many modern games.
Yet, as part of the moe culture, many male players also seem to like the looks of girls with some visible features from cats. However, the step from admiring someone because they're cute to fetishizing is a small one.
Anthropomorphic characters and the fetish around them is still going strong - but this is a venture into the furry community - as stated above in the woren paragraph, usually, catgirls are very much human with just a few feline traits.
In any case, their appearance in modern games is absolutely a cliché that many japanese games like to celebrate. There are only few RPGs from Japan where cat girls aren't present.