I wonder if the trope is so popular because it validates a certain kind of intensity. In real life, being that competitive and obsessive can be seen as unattractive or unhealthy. But in these stories, that same drive is the very thing that makes them perfect for each other. They find the one person who matches their intensity, who isn’t intimidated by it, but is ignited by it.
It’s a fantasy of being fully seen in your ambition and your intellect, and having that recognized as desirable, not off-putting. The rivalry is just the language their particular kind of passion speaks first.