Final answer:
The six-month-old puppy examined by Mateo is most likely suffering from sarcoptic mange, which is a highly contagious mite infestation that causes intense itching and hair loss.
Step-by-step explanation:
Based on Mateo's observations and the course of action he recommends, the six-month-old puppy most likely has sarcoptic mange.
This condition, also known as canine scabies, is caused by the Sarcoptes scabiei mite. Sarcoptic mange is highly contagious, which justifies the vet's advice to separate the infected puppy from other pets.
The mites cause intense itching and hair loss due to hypersensitivity reactions to the mites' feces and the physical tunneling in the skin. Observing the skin scraping under a microscope and prescribing a medicated bath and medication align with the treatment approach for sarcoptic mange.