"Do you mind, I'm trying to get some beauty sleep"
Similarly, a cat's unique ability to purr does often mean they are happy or enjoying affection. However, it can also be a display of anxiety. Look for other signs such as their posture, or if their ears are tucked back - this could be a display of distress despite their purring.
Forget the childhood storybooks of offering a saucer of milk to a cat too. Felines can become lactose intolerant resulting in diarrhoea and because of the high calorie content.
Feeding a cat milk can lead to obesity too so be sure to only feed cats very limited amounts or simply stick to water.