Honestly, I would be surprised if that "obsession" has been around for nearly all of humanity's existence. If you're a hunter-gatherer (or even just hunt or gather to add variety to your diet and for medicine), being able to classify the biological world in some detail is an essential survival skill.
...or at least a very simplified model that breaks down as you move towards simpler and simpler forms of life. That's because horizontal gene transfer is a thing, and when you go all the way back to prokaryotes, there's so much of it going on that any "tree of life" is a statistical picture that's valid only for a select genetic marker that's used to trace dependencies. Pick a different marker, you'll get an entirely different prokaryotic tree of life.
Life ultimately forms a graph
Tangential: it would be helpful if schools explained to kids that all classifications are arbitrary, in biology and otherwise. Their value is in being useful, not correct to some magic metadata tags attached to things.
In context of school biology, I wish we spent more time approaching the classification like: "See that thing in the picture? We classify it as X, because it has $property; subtype Y, because it has $different-property; ... subtype ABC, because initially biologists thought this is different from BCD; now we know it isn't, but the classification remains for now". It would all feel much less boring this way.