Yes, it could be fungal - particularly since you notice odor. Many times, fungal infections have a kind of sour, yeasty smell, or just makes your one dog smell like about 10 dirty dogs! Meds and medicated shampoo can clear that up. I have one dog who gets doggy athlete's foot - fungus. She has really thick toe pads with lots of hair, so moisture tends to hang around in her feet, and dogs have sweat glands in their feet too. Once the fungus gets going, it irritates the skin, so she licks contantly. That brings on a vicious cycle of dampness and continuing the great environment for the fungus. Since it's only her feet involved, I squirt Dermachlor Flush Plus on them and rub it in between her toes. After a couple applications, no more trouble for several weeks to months. It usually is worse in the summer months.
Dermachlor is available from your vet, and contains lidocaine. It's an antiseptic solution that discourages licking as well. You might ask your vet if this product would be helpful for your dog.