It's kind of hard to adress a question that asks, did anything like this exist anywhere over the past 1300 years. But I will try to adress its premises, and in which senses such groups have, and haven't existed.

So we must consider exactly what "unite" means. If you mean claiming to supersede the older faiths and containing a new revealed truth that apply to all adherents of Abrahamitic religions, then yes, there have been many such movements, most obviously Islam itself. Others include Padishah Akbar's Din-i Ilah, Ahmadiyya, Baha'i... there must be countless ones.

But if you mean something like, "all three branches of the Abrahamitic faith are equally true, and are really the same faith in three forms", then I can think of no such movement (though I'm sure someone, somewhere had this idea at some point), at least not one specific to the Abrahamitic religions. I can think of a pretty good reason for this: it would make no theological sense for any of the groups to accept this notion of a tripartite greater truth. The reason for this is that at the very least, Christianity and Islam both attempt to monopolize this type of "greater truth" (the roots of which we can find probably both in Jewish apocalypticism and Greek philosophy).

To convert to Islam, you have to with your entire heart exclaim the Shahada: "There is no god but God, and Muhammad is the messenger of God."

To become a Nicene Christian, you need to accept a variety of notions relating to Jesus: That he was the only son of God, wholly god and man, that he died on the cross, that he rose on the third day, and that his death (and only his death) can redeem (some of) mankind.

There is simply no way to within orthodox Christianity or Islam, to reconcile the statuses of Jesus and Muhammad. Weakening one of their statuses leaves you with something that isn't Christianity or Islam, because their entire theologies rest on notions like the revelatory and final nature of the Qur'an, and the weight of Christ's sacrifice. Judaism has fewer conflicts like this (it is certainly largely compatible with other ethno-religions), but Jews would still have to accept a lessening of their status if they were to accept either Christianity or Islam as true.

As for the Trinity, I know of no such similarity being drawn. Trinities tend to pop up when Christians demonize other religions (Baphomet-Termagant-Apollo, anyone?) or when Christanity is being demonized or derided (e.g. the alleged God-Jesus-Mary trinity). It doesn't really make sense for either someone from a Christian framework to be inclusive of Islam, or for someone from an Islamic background to structure their movement around a Christian concept. And again, the Trinity in its familiar version is one of those really peculiar theological ideas of Christianity (like e.g. the nature of Jesus) that were cemented as a result of settling many, many, many complex theological debates. It's not as simple as saying, "Oh, it's the same thing in three forms, just like the Trinity!" (in fact, I think many Christians might even be slightly offended by such a comparison), and it would actually be really complicated to draw up a parallel that holds and provides something meaningful.

No-one has the recipe for a religious movement being succesful, but a movement that tried to draw on the idea of a greater truth unifying the three Abrahamitic faith would very quickly run into the issue that such greater truth is already monopolized by two of them.