Misaka really loves this frog. Where does this frog come from? Does it exist both in anime and in real life (I feel like I've seen this frog somewhere before)?
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It is very possible that you have seen it several times in the anime ( even in the game), it is a gekota
Gekota (ゲコ太 Gekota?) is a popular mascot found in merchandise marketed to children in the Toaru Majutsu no Index universe. A Gekota fan is called a Gekoer. Misaka Mikoto is a huge fan of the franchise it belongs to, and much to her consternation gets her labeled as someone who is childish.
And without a doubt there is also real life merchandise, and allot of look alike products, with a few samples bellow.