A villa-style mansion can exist, but it must meet the other qualifications of a mansion. It must not only exceed a certain square footage - which can vary based on where it is, In New York City, 5,000 square feet might qualify, because there aren't a lot of residences that large. Anyone with the resources may have maintained a large residence in the city where they worked, but would have built their mansion outside the city where land was available and not as expensive.
But in the areas around Atlanta or Houston, a 5,000 square foot house might be found in clusters in the suburbs. To qualify as a mansion, it has to truly stand out from what's in the area.
This is not only just square footage, but in the quality of construction and in the amenities offered. Mansions have special rooms or areas designed for specific purposes, and in greater quantity than just your large, upscale house.
Villa is just a style. Therefore, a villa-style could be a mansion.