Both words work here and are interchanged. 'Rental' mainly for the type you pick up at airports and 'hire' for weddings, graduations etc. which come with a driver so I concede the point
Language differences can be fun and frustrating. I lived in the US for a few years and can translate pretty well on the fly. On my first visit 25 years ago I was in a diner and asked for extra serviettes (napkins) for the table. The waitress was very perplexed. We figured out later she thought I were saying soviets and was being asked to bring some more Russians to the table!