>>50241994Initially, it was the novelty of a Pokemon that had multiple evolutions, all of different types. It was prime "trade with your friends to get 'em all" material. And on another note furries, but they didn't REALLY start until GS introduced Umbreon. Ever since, the line saw a big popularity boost and its flexible nature warranted Eevee's prominence as a reoccurring Pokemon in the anime.
Since then it's been a gradual climb to favoritism with another Eeveelution Pair in Leafeon and Glaceon in DP, and then Sylveon became the poster child of Fairy Types in XY, which massively boosted the line due to Pokemon's resurging nature in the west from making the jump to 3D.
The real clicker for Eevee becoming a full-time shillmon that ascended past its own purpose of being an Evolution Pokemon was the debut of the Project Eevee marketing/merch campaign back in Late November 2017, which wound up culminating with LGPE, thereby making it an eternal shillmon.