Wait, what? No it's not. Not even remotely close. There is literally nothing about the bard that relates to European animism (or any regional animism). Animism, on a broad level, is the belief that spirits inhibit everything, from rocks to plants to animals to everything else. Even if you look at one of the more popular European animist beliefs, the fylgiur (Norse spirit animals), it's all about the animal spirits helping and guiding humans and the person having an intimate relationship with their animal spirit, where one affects the other. I'd posit the D&D bard is the opposite of animism, because the bard is all about people only. All of the abilities and powers are human centric. I can't think of a single bardic ability in D&D that centers around using mineral, plant, or animal spirits to help the party.