It was fairly widely available in the 19th century as mass manufacturing started to branch out from military materiel and animal rights started to become an issue within society.
It wasn't, however, used heavily for furniture until more recently. If it was described as leather in the 1940's, it was probably actual leather. As noted by KillingTime, the US had a huge cattle industry and there was no significant shortage of leather and it was reasonably cheap. Leather is only relatively expensive nowadays due to the massive increase of faux leather and the consequent reduced production of actual leather, and the perceived "luxury" tag attached to "real" leather. Even so, real leather is still prominent in furniture etc, where that perceived additional quality still counts for a lot.
I have one item of faux leather furniture in my house, and the rest (and my car seats) are all real leather.
It's not impossible the furniture in a motel was faux leather in the 1940s, but it's far more likely to be real.