Idea Factory/Compile Heart... :P j/k
Honestly, I think Square's biggest rival is themselves. So many strange and absent-minded decisions, especially in the 360/PS3 era. Thankfully they seem to have a bit of their mojo back recently (Octopath, DQ11, DQ Builders, and Nier Automata were some of my recent favorites, and I'm loving how they've been showing SaGa a lot of tlc lately).
I was trying to think of western RPG devs, but out of the ones that come to mind, we have Bioware (remember in the 360/PS3 era people were calling them "the new Square"? lmao) which totally floundered, and both Bethesda/CD Projekt whose release schedules aren't nearly as stacked as SE, have only one or two big RPGs, and don't really have the 'legacy' IPs to pad their release schedule. (though I'd still love a Witcher 1 remake)
So I guess that really leaves Nintendo. They're the only other company that puts out a variety of big, successful RPGs on a regular basis - Pokemon, Fire Emblem, Xenoblade, the Mario RPGs... and even some strange one-offs like Tomato Adventure. It's a stretch but you could even consider Ring Fit and Astral Chain RPGs. Kinda funny how this was the company whose former and longest-serving president once claimed it was a genre for depressed gamers in dark rooms.
There are some smaller RPG devs that have big franchises and a steady release schedule - Falcom and Gust for example - and even though their games are praised they're still a bit "niche" compared to Square and Nintendo.