I'm DMing a campaign with a druid player who has been studying dragons and snakes her whole life. Does this allow her to shapeshift to them or does their magicalness make them not "animals"?
No, by default. Your average Dungeon World dragon is an intelligent being like (or moreso) any human and doesn't count as an animal. The Druid is very much about nature and natural animals, not magical beasts or intelligent species.
But, that's by default. In Dungeon World, fiction rules all and in your group's Dungeon World you may have established that dragons are animals. Then it's fine and fitting that the Druid would know them as they know hawks and bears.
But if you haven't established that, say "no" to shapeshifting into a dragon—it would just cheapen the whole fictional underpinning of the move, and generating good fiction is the only reason the moves exist.
If, in the end, you do decide dragons are legit animals, I have one piece of advice: when the Druid misses on a shapeshift roll, make your move very hard. That's how the immense power of the move is balanced—riskier the form, the worse the consequences of a miss. If you pull your punches there, you'll end up with the Druid stealing everyone's spotlight and be a sad GM.
As always with Dungeon World, the answer I'd advocate is: it depends on the fiction and what your group decides that the world should be like.
The trigger for Studied Essence only says 'spend time in contemplation of an animal spirit'. That is vague enough so that your group will have to decide together whether that trigger is met or not. The requirements will probably differ for a squirrel and a dragon.
Is your world one full of magic, where magic creatures dwell abundantly in the homelands of your druid? If so, the Druid may be able to Shapeshift into the native magic species just as well as into the mundane ones.
Or is your world low on magic, where magic creatures are an enormously rare or even unique sight? If so, the Druid may only be able to Shapeshift into the common mundane species.
To address your example directly: in such a world, Dragons might be so rare a species that studying them even for a lifetime might not be enough to Shapeshift into their true form.
To Shapeshift into a unique magic creature, I'd refer to the Doppelgänger's Dance move.
Some things are more than animals.
By "snakes" I'm assuming you're talking about, like, winged snakes that spit lightning or something. Regular ol' vipers and constrictors and such are fine.
In general a Druid has access to three tiers of creatures: starting, Thing-Talker, and World-Talker. If there's something that lives in the Druid's native soil or that they're trying to study the essence of that you believe is more than an animal, think about which of these other tiers it might live on.
Just for example, you might say that a winged snake is also part cloud, and needs Thing-Talker to fully comprehend, while a dragon contains an appreciable portion of pure flame and can't be properly understood without World-Talker.