Rabbits can purr too, for different reasons and with a different mechanism.
One source says that rabbits purr when they are content, for affection, healing and pleasure. The Kid's Britannica says "Other than loud screams when frightened or caught by a predator, the only auditory signal known for most species is a loud foot thump made to indicate alarm or aggression", so I'm not sure about that source. Rabbits may have as diverse reasons to purr as cats.
There seems to be a bit of controversy about how cats purr, but it does involve their vocal cords. Rabbits purr by gently rubbing their teeth together. This is called bruxism.
We had a pet rabbit who would purr every day when I let her out of her cage. She was lonely and happy to be out with me and the two cats. The cats were the reason she couldn't be out all the time. They were polite to her but kept letting me know that they would really like to chase her. Our rabbit was a dwarf species, so she was quite small.
Her purr sounded more like a hum, a warm note. She did not like to be patted much, but she liked to snuggle if I sat on the floor and she was quite friendly to the cats.
People can grind their teeth, but bruxism is more about stress in humans. Crows tap their beaks together, which is like bruxism, but doesn't sound like a purr. Other animals purr, including the big cats and badgers. I don't find much information about whether hares purr.
https://perfectrabbit.com/rabbit-purr/
https://faunafacts.com/animals/examples-of-animals-that-purr/
https://kids.britannica.com/students/article/rabbit-and-hare/276627