What is that frog toy that Misaka loves? 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)?
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.