is it true a 2013 Kia soul does not have a spare tire?

Yes, that's correct. It has an inflator kit, but no spare tire. This seems to be the trend for several automakers as a means to reduce weight and improve MPG. I had a newer Mazda MX-5 Miata that had no spare tire, so I installed run flat tires. which you could also do if you bought the Kia Soul. Here's a list of cars without a spare from Consumer Reports: http://www.consumerreports.org/cro/news/2014/08/your-next-car-may-not-have-a-spare-tire/index.htm

My 2015 Soul ! did not come with a spare tire either. It's got the inflator kit as Racer-X mentioned. Around town I'm not worried, but it was a minor concern on my summer road trip.

Just found out after owning my soul for 16 months that it has no spare and no jack. Seems the seller (Carvana) would have mentioned this. I have been driving for almost forty years and have never before bought a car that does not have a spare. Just needed to jack up the car to get the drain nut off of the oil tank which is on so tight that I can't turn it. Another first.