I just finished a reread of 'The Atlas Six' last week, and honestly, the magical rivalry in that one feels more like cutthroat academia than classic schoolyard stuff. It's less about sports competitions and more about secret societies, shifting alliances, and power grabs within this elite magical institution. The politics are internal, psychological, and deeply personal.
For a more traditional 'rivals to maybe more' vibe within a magical school structure, I'd point you toward 'A Deadly Education' by Naomi Novik. The Scholomance itself is a character, and the politics are about survival and resource scarcity, which creates fierce, pragmatic rivalries among the students. You get alliances of convenience and the constant tension of who you can actually trust when everyone's trying not to get eaten.
A lot of the popular 'chosen one' school series gloss over the bureaucratic infighting, but these two really dig into the systemic pressures that make rivalries inevitable, not just personal.