I wouldn't say it's synonymous with "Summons." The term "Espers" is used with similar frequency, appearing in VI (the first time summons were given a unique name!), XII, Tactics' retranslation and all of its sequels, Airborne Brigade, Record Keeper, Brave Exvius, and Chocobo Racing. Even Dissidia 012 slips back and forth between the two terms. And, more importantly, with regards to mainline Final Fantasy games, no special name was given to any summon for six games (with IV only calling its summons "Eidolons" in its remake) and four of the games chose their own game-specific terms (GFs, Aeons, Avatars, and Primals for VIII, X, XI, and XIV, respectively).
Basically, "Eidolon" is not a safe translation for a Japanese term the way that "Moogle" is a safe translation for "Moguri." It's a specific localization choice made by some FF teams and rejected by others. I find it difficult to believe that a professional translator would decide it was a good idea to call summons "Eidolons" without being told to do so.