In multiplayer, all players will see the same custom names.

Why is that? Can it be changed at some point?

Modding limitation. This mod works by changing the display name of qualities, which is something shared by all clients.
If you want to change quality names just for yourself, without a mod, you can change the names in the translation file here: factorio/data/quality/locale/<your-language>/quality.cfg. That's not something mods can touch unfortunately.

unfortunate, thanks for the reply.
That's what I thought. Would have been helpful for a mod I'm making.

In theory, one could fiddle around with localizations. If each player sets a different language, mods can assign different localized names per language. So, effectivly, you could replace some language you dont need by a copy of english with different names for the quallities. This is not very practicle, but if a mod can add new languages, you could make this work. So, "English, legendary quality" , "english, flawless quality" and "english, smiley quality" could be defined as different languages.

You can manually achive this by going to C:\Program Files (x86)\Steam\steamapps\common\Factorio\data\quality\locale\en [If installed via steam] and editing the localization files. They will probably be overwritten by game updates (I dont know how often). This changes the names displayed to you without changing them for every player. It also should not interfere with achievments.