When a doctor or nurse examines testicles the technique is not a caress as much as it is a feeling of what lies beneath the scrotal skin. She is looking, or rather feeling, for lumps, sore spots, and masses, and checking that the cords and testes are the "right" size and position. It is done behind the testes and above them, close to the body.
When a hernia check is being done again it is not caressing the testicles, but rather a finger that is poked up into the inguinal canal (through which the teste descended when the boy was young), of which there are two, one on either side of the root of the penis.
When the testicles are being examined for skin cancer (rather than cancer of the testes themselves), sometimes the skin on the scrotum is pulled taut, if need be, so as to see the surface more completely. Also, if there is much pubic hair, she may need to slightly "caress" the skin in order to see in and among the hairs.
Finally, as the doctor in the video explains, she needs to cup the balls in her hand, checking for size, similarity and weight and also for how easily they hang and move in the scrotum (and in the case of one doctor I had, because she liked the way it felt...or at least I think that was why she did it for such a prolonged time, not that I ever complained, mind you!)