Satanism cannot be properly described as an Abrahamic religion. This is true even though most forms of Satanism share a good deal of DNA with the Abrahamic faiths, including a great deal of the mythos. To properly be called Abrahamic, a religion must be covered by the covenant that God made with Abraham. While this is true of Judaism, Christianity and Islam, it is certainly not true of Satanists, who are breakers of any and all covenants with the God of those religions.
Satanism has been described by some detractors as “just Christianity,” but that’s about as nonsensical, inaccurate and insulting to both faiths as describing Christianity as “just Judaism” or Islam as “just Christianity.” To be “Christian” one has to accept Christ as savior. That is the bare minimum of the definition. Satanists obviously do not meet that definition.
I also would shy away from calling Satanism “Neo-pagan.” Neo-paganism strives to recapture something of pre-Christian faiths. Satanism can only be understood as post-Christian. It is a modern phenomenon (particularly in the sense of modernity as beginning around the 1500’s. See Children of Lucifer by Ruben Van Luijk for a deeper understanding of why I say this). Neo-paganism is also often more explicitly Earth-centered than Satanism. This is not to say that ecological concerns and nature-worship have no place in Satanism– they certainly do!– but they are not as fundamental, as central, as they have been in many Neo-pagan movements.
Satanism is strongly influenced by the Western Esoteric tradition, but even within that it is an aberration, since much of Western Esotericism has been Abrahamic (and specifically Judeo-Christian, and this may be one time when that term can ACTUALLY be used accurately*) in nature. I also find it difficult to call Satanism truly “occult,” since “occult” has connotations of secrecy, and much of Satanism today operates fairly overtly and involves few hidden teachings. (There are occult orders within Satanism, of course, but Satanism is not always occult per se.)
Thus I would classify Satanism either broadly as post-Christian, or as a category unto itself. Certainly “Satanism” is a broad and diverse enough term to justify being its own category. You have your atheistic Satanists– LaVeyans, and The Satanic Temple, and others. You have your theistic Satanists and Devil Worshipers. You have your Luciferians, your Setians. You have your Demonolators and Thelemic Satanists and, of course, the fascist Satanists who can fuck off like the Nazi punks they are. The only thing all of these groups have in common is that they say “Satan” like it’s a good thing, that they revere, worship, work with, admire, or aspire to the image of the Adversary.
In closing, Satanism is a unique development occurring in a unique time– this time. I cannot say with certainty when real Satanism began– scholarship on this question is developing, and new findings do suggest that it is older than LaVey, but probably only by a couple centuries at most (see Van Luijk again). (I personally consider Crowley to have been the first genuine Satanic leader, but I know a lot of Thelemites who would violently disagree, not to mention the entirety of CoS.)
What I can say with complete assurance is that there are more Satanists today than there have ever been before, that the advent of the internet has been absolutely crucial to the proliferation of Satanism, and that this movement is only growing. Serious academic research on Satanism is a rather new field, with a lot of ground-breaking work coming out only in the last five years or so.
This is our time. We are a new religious movement that is gaining serious momentum. It is inevitable that we will be studied further, and study means attempts at classification. That is why I propose Satanism as its own, individual and unique category, in lieu of ungainly attempts to lump it with Neo-paganism, Abrahamic faiths, occultism, or Western Esotericism.
*I used the term “Judeo-Christian” to refer to Western Esotericism because the bulk of it is Christian in ideology but borrows or steals prolifically from Jewish Kabbalah, and loves employing (usually garbled) Hebrew as a magical language. Islamic alchemy and esotericism are influences on Western Esotericism as well, but markedly less so than Jewish esotericism. Most of the time “Judeo-Christian” is a stupid term used when people actually just mean either “Christian” or “Abrahamic,” but in this case I defend my useage because Western Esotericism is genuinely a combination mainly of Judaism and Christianity.