Just some quick notes from my perspective...
Plantkin are typically called "phytanthropes" and also have a symbol, akin to the Theta-Delta.
From my understanding (and I'm not speaking for plantkin, not being one) they are (like therians) a subset of otherkin but having their own terminology. Being plants rather than animals, excludes them from the definition of therianthrope -- hence phytanthrope. I don't know if there's a general consensus on this in the modern community as plantkin are pretty rare.
I agree with @WolfVanZant above, having read some accounts (such as that from @House_of_chimeras
, they have a very good description) -- I would also argue that it is every bit as real a phenomenon as therianthropy. Perhaps, a -thrope prefix exists that covers everything? Allothropy?