According to the former website Musica Viva—that sadly went offline and is also not available through Internet Archives— saxophone-shaped blow accordions were a step-up, or fancier version of the simple design that was the most common.
These blow accordions were early variations of melodicas. However, rather than using a piano-style keyboard, these blow accordions used keys similar to a button accordion.
Most commonly blow accordions had 10 keys with 2 bass/chord buttons, but other variations can be found as well. Variations included instruments with more or less buttons. Those with more buttons had 2 rows of keys.
According to Musica Viva, the most important differences between the modern melodica and these vintage blow accordions, was that the vintage version had a double action—producing notes when air was drawn both in and out—and that it had bass/chord buttons, like a melodeon.
I have a number of different blow accordions in my gallery. Various manufacturers throughout Europe made them during the first half of the 20th century.