These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Awesome Art: The concept and promotional art use a strange, very unique, very colorful style that looks wonderful. The sprites are also some of the most popular in the Mario franchise, to the point that it's popular to customize them.
Though they caught on and made him the main villain in the third game.
While Popple came nowhere near Fawful levels of popularity, he was still the second most popular character from the first game and made a return for Dream Team.
Now that Fawful has been Killed Off For Real, Dream Team players have found the Massif Bros as two more characters with hilarious speech patterns, between their Russian accents and their constant mention of BEEF!
It's also common to make sprites styled like the Mario and Luigi series sprites, from power-ups (Tanooki Mario, Shell Mario), to characters (Wario and Waluigi).
The Woobie: Luigi, definitely. This scene and this scene from Superstar Saga and Partners in Time respectively, increase Luigi's woobieness when he breaks down crying in both of them.
Woolseyism: The games are known for diverting drastically from the Japanese scripts whenever Rule of Funny comes into play. The Hammer Bros. in the second game in particular definitely did not speak in 1337 in the original.
Fawful is interesting here, because he talks in a different funny way in every single translation. He adds Ru to the end of every sentence (repeatedly) in Japanese, has mangled Engrish in English, a strong accent in French, stutters in German, speaks the same in American Spanish versions, and speaks in refined Spanish with set phrases in the European Spanish versions. He speaks in metaphors in multiple versions.