yes uneven tire wear is not uncommon on lifttrucks
especially if the operators in the past had poor driving practices and did alot of directional changes without using the brake allowing the tires to spin. Being a limited slip differential usually one tire will spin and most of the time it will favor one side over the other. Is it possible that an odd tire was installed on the lift prior to you getting it? sure it is. Can this cause driving anomolies like you describe? i suppose it can to a degree. I've been on lifts with tires worn like this that drove fine but while driving the lift would bob back and forth due to the uneven tires... its a tell tell sign the lift has been driven abusively. So yes... put tires on the lift... both tires... never replace a single tire as it creates unlevel travel which will be translated into uneven forks....
It makes operating the lift difficult when entering pallets...