I've never performed any rigorous tests using different types of mounts, so can't give you any first-hand experience here. However, birch sap contains methyl salicylate (which is what gives things their "wintergreen" flavor), and it's been used in poultices for wounds and to help reduce swelling. Birch wood might, therefore, have some anti-bacterial / anti-fungal properties, but that's a big "might."
Does the grower who made this claim actually buy and pay to ship birch wood mounts to his location, or is there rather an extensive forest of birch nearby through which he can hike and cut mounts for his plants, there being no other type of hardwood as cheaply and easily available?
There's certainly nothing *wrong* with using birch, and it may actually provide some benefits that other hardwoods don't, but I'd be more willing to bet that there just happens to be a lot of it around, a situation not unlike a cereal company touting the benefits of breakfast, in general, and their product for breakfast, in particular.
"Grain of salt" is needed here....