He is basing his diagnosis on generally accepted knowledge...primarily that young toddlers aren't susceptible to strep throat. He'll also take into account your child's age, whether or not he is in childcare, and the pattern of the emerging symptoms. HFM is common this time of year and there are a whole host of symptoms that may be more or less severe in individual children. I'd say its not strep just based on the fact that he is eating and drinking fine.
My daughter has had HFM twice (and she's only 3.5)...most recently just a few weeks ago. It is caused by a several viruses...not just one, so repeat events are possible. It was milder this time only with a significant fever and a red, patchy tongue. The first time, there was only a few days that she didn't really eat...and she was completely breastfed at that point so it was a different ballgame.
The sores generally appear/worsen several days after the onset of the fever--and he may not get them at all on his hands or feet (and that may just be more of a rash than sores). His throat would also generally have pockets of pus with strep, but just red and irritated usually with HFM.
A quick strep culture has a high false reading rate...by the time they do a full culture, you'll already know if he is getting better or worse and the doctor will re-evaluate. I am happy that there is a doctor out there that isn't still just throwing antibiotics at everything.