I'm surprised no one has replied yet.
At any rate, here's my take:
Let me preface my comments by saying that, of course, no one is
required to either like or respect a particular piece of great literature or "one of the classics", if you will -- certainly you'd be in a rather large company if you were to question
Silas Marner's standing, for instance -- but, as you asked how this one has stood "the test of time", well, here are some of my responses:
First, the objections you raise above (his apparent self-teaching and such) were raised by reviewers upon the novel's original publication and have garnered various responses over the years, from Sir Walter Scott on. On a strictly literal level there are various reasons which are hinted at within the book: the creature's almost preternatural intelligence and cunning, for example, smack of a diabolical spirit to begin with; a possibility that lies within the very epistemological nature of the novel itself: is this being good or evil, is it soulless, is it (being constructed from the dead) inhabited by either the soul of someone who has died or an infernal spirit? In which case, such feats would be very much in the nature of any of these according to the folklore common to such. Mary Shelley wisely maintains silence on the issue not because (I suspect, given the fact that this novel was not only the product of long gestation and considerable thought, but also revised and rewritten in part with suggestions from her husband Percy and revised again with newer editions later in her life; also based on the care with which she constructed all her other works, given the evidence of her various journals, diaries, and letters, including those on this one) she didn't think this aspect through, but because she wished to maintain ambiguity on this point as well as on one which I will raise later.
Recall that this was a time of transition in literature, where the rationalism of the earlier eighteenth century was now being challenged not only by the Gothic reaction toward emotion, but also by its (partial) offspring, the Romantic movement, as well as the burgeoning tales of the fantastic and terrible coming out of Germany (on which more later) -- something the novel itself reflects, being poised between the superstitions of the past (then seeing revival) and the rationalism of the post-Enlightenment and burgeoning scientific, even mechanistic, era (a question the novel also delves into as well).
Nonetheless, the beliefs surrounding such beings were still quite maintain ambiguity on this point as well as on one which I will raise later.
Nonetheless, the beliefs surrounding such beings were still quite widely known, and therefore not the sort of thing needing going into at this point, as the majority of readers would be quite aware of them and accept them as part of the worldview of the tale. Only later, as the more clinically scientific tone began to enter into the fantastic tale, would such things be delved into and explicitly stated.
Second, on a different level, the being "shadowing" Frankenstein is because it literally
is his shadow: In the same way that Hyde is Jekyll's, so is Victor's creation his shadow self. Where Hyde, originally dwarfish and wizened, gradually becomes less so, to the point where he is described as not actually deformed in any particular yet giving the
impression of ugliness and deformity, also becoming stronger than Jekyll -- enough so to emerge with or without the potion or the will of the doctor -- as the latter gives more of himself to the former; so Victor's creation literally overshadows its maker because of his obsessions and the fact he pursues them without regard to either his ties to others or the impact of his actions on them. It may be significant here that the only other view we have of the creature is through Walton (and his crew), who himself is engaged in a similar scientific pursuit. Note that the only
clear view he has of the creature is once he, having already heard Frankenstein's tale, is on the verge of continuing in his footsteps (so to speak) -- at which point he encounters the being face to face, which may be seen as a portent of his
own shadow growing. Other than that, we only have Victor's testimony to the thing's existence -- once again indicating that it is his double (just as Walton is, in a sense, of Victor before his "fall", so to speak).
In other words, the novel is interpretable in a number of ways: literally, metaphorically, mythically, within the various traditions to which it belongs (or which it in turn fostered), etc. Which brings us to my main point: The novel survives because it has stimulated imaginations for nearly two hundred years (and continues to do so); because it tackles fundamental questions about life, religion, human nature, the existence of justice, of god, etc., exploring them through dramatic action; because of the very mythic nature of the novel, straddling, as it does, both the scientific/rational and the mystical/irrational.
I mentioned the tales of the German Romantics (and, by extension, the
Schauerromantik), which so influenced the Gothic tales and the Romantics such as Mary's husband and Byron (among others)... and Mary herself. These, too, were riddled (quite deliberately) with the sort of "logical inconsistencies" which appear in spots in Mary's novel; yet they, too, work both because of the quality of myth they contain (and the resonance with emotions surrounding such themes and tropes) and because, at their best (like Frankenstein) this gives them something of the feeling of nightmare, with its logical distortions which yet are bizarrely fitting.
Chiefly, though, I suspect it continues to succeed with the
popular (rather than the critical) reader because it appeals to the imagination and the emotions in its exploration of the questions of life and death and the ambiguous nature of the creature itself, which may be viewed as a fiend without compare, or being potentially good but, scorned not only by those around it but by its creator, turns into an enemy of all -- eliciting our sympathy by its (apparent) struggles to maintain the good within it and echoing the feelings that most of us have at one time or another of being the Outsider....