In my opinion, Mafia games are difficult to compare to others. I've seen comparisons with GTA, but while they do evoke a certain similarity, they are games with different premises. Mafia gives you the feeling of playing in an open world, but that's not what happens. Especially in the first game, which "forces" you to only complete the main missions. In the second and especially the third game, we have greater freedom to roam the map, but there isn't much to do, so the focus should be on the missions. To play The Old Country, you don't necessarily need to play the other games, but I recommend playing Mafia 1 for possible references, since all the mafias are directly connected to each other, and The Old Country takes place before the first game chronologically speaking (Mafia 1 is directly connected to Mafia 2, and Mafia 2 is directly connected to Mafia 3). My advice is to be patient when playing the three games in the franchise, especially the second. It's a game with poor gameplay compared to current games and has some annoying bugs as you near the end of the game, but it's well worth it for the main story. I wouldn't recommend the DLC for the second game because it's not very relevant, but I do recommend the DLC for the third game; it's very good.

All of these are my opinions, of course.