Almost every word written in katakana is a loanword, and most of them come from English. There are many different words in Japanese that can be translated to "damage"/"harm"/"injury" etc., but I guess none of them carry the exact connotations that ダメージ does. I've noticed that loan words are very common in Japanese video games, sometimes for words that to me appear to have an exact match in Japanese. I don't know why they do this. There are also some writers that make an effort to avoid loan words and use more traditional Japanese, but this is not so common.

An example which I find amusing is お金ゲット！(okane getto, money get). There are perfectly valid Japanese alternatives to Getto, and to an English speaker, this sentence doesn't even make sense. That's not how "Get" is used in English grammar at all. But in Japanese it's kind of a playful way of saying you acquired something.