the thing about Revolutionary Girl Utena is that "Anthy's role in the illusory system of Ohtori gives her godlike magic powers" is more or less equivalent to "Anthy's role in the narrative of Ohtori gives her godlike magic powers"
— I don't really hold much truck with the idea that Anthy, Dios, and Akio are some biologically or ontologically distinct class of entity, and that's what allows them to work these marvels; I think that takes the Tale of the Rose far too literally, against the overall attitude the story articulates about stories, and also is boring —
so the thesis that "Nanami is so plagued by animals because Anthy is using her magic powers to torment Nanami" doesn't quite sit right with me.
don't get me wrong, I do think Anthy sometimes does things out of spite or sometimes enjoys her role in the various schemes she is tasked with working out (and can you blame her, given what that role asks of her? [pointing a knife] No, You Cannot). and certainly Chuchu did sneak that egg into Nanami's bed, and Anthy did buy Nanami the cowbell
but "every animal involved in Nanami's tragicomedy was put there by Anthy's magic" just doesn't seem like Anthy's style. Anthy is subtle, and she ensures her own plausible deniability! the animal pranks we DO get to see Anthy doing to Nanami (sending her a fancy cowbell, sneaking an egg into her bed, giving Touga a kitten) are, one, pretty subtle and, two... arguably very mundane!
even though those situations quickly spiral out of control and become very complicated, they start with planting a strange idea in Nanami's head and letting Nanami do what she does best.
you know whose style DOES fit with being stalked by a horde of elephants, or menaced by a set-loose boxing kangaroo, or having the egg you 'laid' turn into an alien spaceship? it's Nanami's!
I won't go so far as to say that Nanami causes all of her own problems, or does all these things to herself. in fact, that's exactly what I also dislike about the "Anthy casting Summon Wild Animal several dozen times in an episode" interpretation of what's happening to Nanami; both Anthy and Nanami are agents and victims within a system, playing a role in a story.
in particular, Anthy and Nanami have the same role in the same story. Ohtori rests (read the Palace Perspective!) on abuse of a little sister by an older brother, on the little sister suffering in place of the older brother. this is what Touga sees in himself that he also sees in Akio; it is what Nanami, who has not-quite-yet assumed or understood the role of fulcrum in this system of abuse, refuses to see in herself or relate to in Anthy.
as it plays out for Nanami, that role centers powerfully on the ambiguous and threatening relationship between humanity and animals, whether being menaced directly by an animal, being cast as an animal (a cow, 'vermin/buzzing insects'), or cast as beneath an animal — when Nanami gives her brother a cat, she finds that what she has given feels more valuable to her brother than she herself does.
she strives constantly to distinguish herself from animals and social lessers, which she identifies with one another — and whether you find that habit of thought intentional and insightful about oppression, or clumsy and troubling, either way it is a striking bit of character writing for a blonde, pale foil to dark-skinned, dark-haired Anthy — but she cannot get it right. this animal/human distinction that she weaponizes is ultimately patriarchal, it can only render men as fully-human. so Nanami cannot prevent 'self' from collapsing into 'livestock,' she cannot identify what it is about her identification with this oppressive system that makes her feel so troublingly animal. her own reproductive capacity, in the sideways aspect of an egg she supposes she has laid; is it normal for a human girl, or does it make her some kind of bestial freak?
when Nanami seeks her older brother's protection, she is menaced by animals (and when Tsuwabuki seeks to be Nanami's beloved older brother, he creates situations that menace her to rescue her from! he, too, has found a role to enact in this story!). when Nanami seeks to be restored to her brother's good graces (by going on an international fetch quest for seasoning), she is menaced by animals.
when Nanami seeks to excel, superlatively, in being the center of attention; to exceed other women in the feminine dominance of being gazed upon (when Juri upstages her at her own party); it is perhaps unsurprising that Anthy's prankish gift should find its way to the spotlight Nanami creates — but Nanami's refusal to acknowledge the abasement such patriarchal forms of value entail is her own. it is her wilful refusal of the obvious that spirals into her turning into a cow, it is the contradictions of her relationship to her brother that meld in a dream into her brother selling her to the butcher and eating her as veal.
when Nanami begins, gradually, to understand with the passage of time and her entry into adolescence that something — even though she does not yet understand what — is wrong, the nightmarish distorted menarche of Nanami's Egg is exactly the panicked and backsliding search for clarity of an isolated, naïve young girl who has no-one to ask about what is happening to her, and receives only rebuke for her sideways attempts to find out.
the 'magic' on display here is exactly the outlandish, fantastical outward expressions of Nanami's own, mundane, internal and interpersonal contradictions. why would that be Anthy's particular magic, rather than Nanami's? or (perhaps more to the point) a magic which does not properly belong to either of them, but to Ohtori's fairy tale in which they play parallel roles?
read the beats in the study group with Nanami, Anthy, Miki, and Utena; three times, Nanami thinks she will expose the bizarre unpleasantness she senses yet does not understand in Anthy, unaware of the hypocrisy she employs — "what kind of girl keeps a snail on her person?" — and each time, she finds a much greater strangeness than she could even imagine to invent.
if Anthy is to be read here as the witch casting spells, why is Nanami the one who is giving the unannounced and underhanded 'gift' (three animals), just as she did one episode prior (the dissolving ballroom dress)? it's only much later in the series that Anthy 'participates' in this exchange of 'gifts' (the cowbell, the egg)
what happens in the study group is only this: Nanami brings her best wickedness, and Anthy shows her a greater wickedness in the same coin. if Anthy is using her 'powers' here, it is only as a rebuke from a greater witch to a lesser witch: I am much, much better than you at your own game.
if Anthy's underhanded gifts to Nanami plays a critical role in both Cownami and Nanami's Egg, it is because Anthy understands something about Nanami. and if Anthy resents Nanami in particular, more than the myriad other people she is asked by the story to play victim to, it is because Nanami's underhanded gift of the dissolving dress shows that Nanami understood something about Anthy. what right does this novice witch have, to remind Anthy of the faces crowding around her at the stable?
also, and the centrality of the following to my reading of Anthy's role in Nanami's Animals Thing TM cannot be overstated: it's both funnier and more devastating if Touga was never 'in' on the egg debacle. when Touga says "Do you know why you and I are able to live together so happily? It is because you aren't the kind of girl who lays eggs," I don't buy for a minute that Touga plotted this out in advance and enlisted Anthy's help so he could eventually deliver this line.
who does Touga model himself after? Akio, who is always playing off unexpected events as 'just another layer of his scheme'. Akio is an improviser, a bookie rigging all outcomes in favor of the house, much more than he is a watchmaker building clockwork. Akio is an improviser, and I honestly just like that scene so much better if Touga is improvising.
he picks up that Nanami is having another of her spirals (which is all the better for him), and that it involves eggs. so he makes a stab in the dark at what could possibly make Nanami spiral even harder.
hot take!: it's not even a particularly lucky or difficult or 'out there' guess at what will hurt her. he's just juxtaposing their 'happy' relationship with the threat of its ending, the 'happy' family and sibling relationship that secures Nanami's place in the world, yet can be destroyed at any moment by... whatever it is about Nanami that is so ineffably strange and shameful. it's always the familial hot-and-cold with Touga, because he knows that's the form of cruelty that always works to dig Nanami deeper.