It looks like they have one significant competitor, New Holland (i.e. Cnh Industrial): https://csimarket.com/stocks/competitionSEG2.php?code=DE, and two smaller that are worth mentioning: Massey Ferguson (Agco), and apparently Emerson (though I couldn't find which brands they own at a quick kagi'ing.)

Boeing actually seems to have more diverse competition, including Lockheed, Northrup, etc (although they're dominant in the commercial airplanes segment, where airbus isn't mentioned, so maybe these stats are US market only): https://csimarket.com/stocks/competitionSEG2.php?code=BA