Hi Vedberg, welcome to the community.
I believe you are talking about The Godfather series, based on the movies. All Mafia games are story driven and I played Godfather myself when I was younger. The focus was to grow your mafia empire by threatening store owners and such. Can’t think of any other games than those.
The Godfather 2 is not much to talk about but The Godfather 1 was a master piece based on Mafia 2 move with actors from the the movies actually voice acting the game characters.
Have a look at: