Asian elephants are Elephas Maximus and are found in Asia like tigers are.
African Elephants belong to the genus Loxodonta and are found in Africa, naturally, with hyenas, zebras, ostriches, and cheetahs.
You say that anacondas and macaws are found in South America, and that seems correct. But who says that the giant snake in the movie is an anaconda instead of an Asian python? Don't claim you can tell that the snake "actor" was actually an anaconda, because the nationality or even species of the "actor" is irrelevant to the plot. The nationality or species of the character is all that is important to the plot.
Just as human actors are often of different nationalities or races than their characters, animal "actors" are often of different species than their characters. But it is the species of the animal character, not the animal "actor", that is important to the plausibility of the story.
A goofy theory to justify the island fauna is the medieval geographical conception of the "Three Indias" that stretched from East Africa to southeast Asia & sometimes China & Japan. In 1165 a famous letter circulated in Europe claiming to be from Prester John, claiming to be the mighty Christian ruler of the Three Indias. So maybe in the fictional universe of Swiss Family Robinson Prester John had his capital on or near the island and brought exotic animals from all of the Three Indias (and maybe also the West India of the Americas) in his realm and the island still had descendants of some of those species. So if they had done some archaeology they might have discovered lost treasures of Prester John.
Or possibly the island was near Skull Island and shared some of its fauna..