It's a mixture of genre's really there are bits where you are running about exploring, fighting bits, minigames, puzzles etc. I suppose overall it might come under Adventure as you said but not really in a straightforward way.
The first one is set in Japan in the 80's and it's about a guy looking for his fathers killer. The end of the first game is sorta inconclusive and in the second game he sets off for Hong Kong still on the same trail. I think they had originally planned a third game set in mainland China but that has yet to materialise.
The main thing about it that draws you in is just the immersive environment, sure the voice acting is a bit cheesy "lets get sweaty!", "i'm looking for some sailors" etc. but it really does feel like you are running around a small Japanese town and all the minigames such as collecting the toys from the capsule machines and various arcade games scattered around really give it that extra something.
For me the atmosphere of the second one never quite matched it, though it was more ambitious in scope with stuff like training at a shaolin? temple and doing various odd jobs around the docks, arm wrestling etc and a longer story.