It was my understanding that no, it did not have splinters, but that someone began marketing it as "splinter free" in the same way I could begin advertising my cereal brand as "asbestos free" and while I'm not outright SAYING the competition contains asbestos, it's implied. And if anyone reads it and chooses my brand because of their assumption, that's just a bonus and not at all my fault.
Using paper specifically for use with toilets was first written about around 1500 years ago, but the casual way it was written about tells me that it was commonly used for some time before that. If I had a guess, probably for at least 150 years, but China has been making paper for almost 2,000 years, so who knows when someone first tried it on their ass.
If normal paper was splinter free, it stands to reason that toilet paper, which must be thinner and less rough by necessity, would also have been splinter free.