It's for correct pronunciation. If there's no two dots on the u letter, french people may pronouce it 'emmo' because an 'a' followed by 'u' is pronounced 'o'.
But there's no need because Emmaüs is so well known in France that everybody knows how to say it.
You can find other examples in words like 'aigü' (acute) where you have to say the u like german ü.
"Aigu" does not take a trema. The feminine form "aigüe" does, and the trema is there because otherwise you'd pronounce it "/ɛɡ/" instead of "/ɛ.ɡy/". It does not change the sound of the "u" at all.
