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Alright, let’s take this from the top.
Mario’s a character, not a show. He’s mostly in video games. So no, Mario is not an anime.
Now then, if you mean the Super Mario Bros. Super Show, and all that jazz, then technically speaking yeah, it’s an anime.
Broadly speaking anime is a term for any fully animated series of pictures making up a show or movie, it’s jusf short for animation and all.
However, strictly speaking the term “anime†refers to these kinds of pictures coming from the Far East. If it’s made and shown here in the states it’s called a cartoon.