Theres a lot of reasons for that IMO. Probably the two biggest are the melee hard-carry meta in DotA and the way the jungle works.

Battlefury, like Tiamat, deals more splash on melee than on range, so melee carries want it (predominant in dota), ranged don't (predominant in league). The reason it's a carry item is because it lets you farm many times faster on heros that don't have wave clearing abilities (like... most carries). If you ever played a decent amount of hard carry in DotA, you will grow to love battlefury for exactly that reason.

As for the jungle, DotA has a stacking mechanic. For those that don't know, monsters spawn in the jungle every minute (x:00 on the game clock), as long as nothing is currently in their spawn box. This can be abused by aggroing the existing monsters and pulling them away from the spawn, allowing a second camp to spawn despite the first one still being alive. Typically a support will do this in their spare time, or the carry can just do it while farming. A carry with Battlefury can then go destroy this "stacked" camp MUCH faster than if they didn't have the item. Even without stacking, several dota jungle camps have more than one "big" creep, so having some multi-target damage is useful.

Disclaimer: Not an expert, just a long time DotA vet; this is my play-group's consensus and may suffer from group think.

TL;DR Battlefury is a farming item for hard carries in DotA for reasons that aren't applicable in LoL.