Mario Party 2 used some mini-games from the first game. These mini-games were usually just changed visually such as Handcar Havoc, Face Lift, Shy Guy Says, Skateboard Scamper, and Tipsy Tourney. There is some that also returned and have similar or the exact same gameplay but with a different name such as Hexagon Heat (Mushroom Mix up), Dungeon Dash (Desert Dash), and Rainbow Run (Tightrope Treachery). Hot Rope Jump also returned but it's a tiny bit different and more improved in my opinion. Toad Bandstand is an interesting example, it's Mario Bandstand but as a 2 vs 2 mini-game and it received a name change.
It's not really a bad thing if you ask me, they brought back some of the best ones from the first Mario Party and improved some others, it made Handcar Havoc a little more tolerable.
Most of the Mario Parties have a mini-game based on Slot Car Derby from the first two games, where the objective is to go laps around a small track usually in some type of vehicle. Water Whirled was my favorite of these types of mini-games. Mario Party 5 has you ice skating instead. I honestly got tired of these kind of mini-games after Mario Party 5. I think Mario Party 6 is one of the few Mario Parties to not have one but I'm not sure. Are any others lacking one?