It's not an AoE so it definitely wouldn't do increased damage to the swarm. I'm uncertain whether it can damage the swarm because single target effects usually don't harm a swarm, though it's unclear whether or not Caustic Blood would be considered a single target effect.
It's unclear...but I lean towards the idea it doesn't work against them.
I'd lean towards "does work", but more from Rule of Cool and RAI acting as tiebreakers than RAW providing a solid answer.
I agree that it probably wouldn't work officially, but I'd probably allow it to anyway, just because it seems like it should. Swarm damage seems to be typeless, even if it would make sense for it to be slashing or piercing in many cases.
It looks to me like most swarm damage is untyped, so it wouldn't trigger the spell in any event.
Beyond that, it seems pretty clear to me that the spell effect is going to target the 'first creature', and targeting the first creature is clearly targeting a specific number of creatures (1). By strict raw, I don't see any other way to rule this.
However, the flavor of this spell is that although it doesn't splash and only will damage a single creature, it is a spray of blood over an area. I would probably have it damage a swarm but not do extra damage in my home games.