While it seems cats don't learn to purr from other cats, I do know of a dog who learned to "purr" from a cat
My old dog Jade (a border collie, lives with my mum), was rescued a year after we got our cat, Ash. They got on fine, though Ash was of course the boss, and she doted on him.
Jade was rescued from a farm at 18 months old, so had never been housetrained. She knew she shouldn't wee in the house, but didn't know how to ask to get out. Ash used to get let out of the livingroom window, which opened onto the garden.
One day, jade got up on the back of the sofa, and pawed at the window !!!!
Around the same time, She stared to "wheeze" whenever we petted or hugged her. Concerned, we took her to the vet, but nothing was wrong. We soon noticed she only did this when happy or being fussed over, and that she was very deliberately making herself wheeze. The penny dropped - she used to sleep with, and cuddle up to, the cat - who would purr.
Jade, like the window incident, was copying Ash! She was "purring"!
She still does it to this day - we have to keep explaining to people who meet her that no, she isn't asthmatic, she just thinks she's a cat!