No, dye means actual dye.
Neetsfoot oil causes leather to darken, but that's just a side effect. It's really a leather conditioner.
I've used neetsfoot on all my bridles, because even though I use a synthetic saddle and neoprene girths, I've always preferred leather bridles.
I've never encountered the bridles staying oily or smelling badly.
You have to make sure to wipe off any excess oil that isn't absorbed into the leather. Neetsfoot does have a smell to it, but I've never found it to be unpleasant.