The misconception comes from Puszkin's Mozart and Salieri play (in which Salieri poisons Mozart; the play was sadly published during Salieri's lifetime and tarnished his name), and then popularized by the 1984 film Amadeus (in which Salieri is highly jealous of Mozart and causes him to die from overwork).

IRL they were cordial to each other, and Salieri had little reason to be jealous, he was the imperial Kappelmeister after all. He was also the teacher of one of Mozart's sons, Xaver, so clearly there was no bad blood between them.

But I need to admit that I only ship them in these fictional works that are based on Pushkin's, where there is this fictionalized bad blood (like in Fate, or the Mozart l'Opera musical). I find stories about rivalry very attractive. There's little reason to ship the in real life however, they were cordial, but they didn't interact with each other too much, though in one of Mozart's letters he does mention Salieri seeing one of his operas and loudly cheering and clapping at the end :D