I recently saw a video on TikTok that claims that Mario actually says "itsumi" instead of "it's a-me". The video claims that since Nintendo is based in Japan, in Japanese, the word "itsumi" means 'superb' or 'super', so really, he was saying "Super Mario" all this time! Amazing! However, this claim is not true. Mario does say "It's a-me" and you haven't been getting his catchphrase wrong.
According to the official voice of Mario, Charles Martinet (Who recently announced his retirement), He's was actually just saying "It's me, Mario!" but it sounds like the character is saying "It's a-me" due to his Italian accent. Furthermore, 'Itsumi' doesn't actually mean 'superb' in Japanese. It's a surname that can mean 'superb, great, and outstanding' but that's as far as the link goes. Some of Mario's other catchphrases include "Mamma Mia!", "Wahoo!", Yiiiiiipeee!", "Let's a-go!" and "Okey-dokey!" and none of them mean anything in Japanese.
So, no. Mario isn't saying "Super Mario" in Japanese.