Is a steam unit for a steam shower considered a continuous load? Does wire need to be sized at 125% of load?

I doubt it. I would be very surprised to find a high-pressure steam tank on such a unit; I expect that they only make steam when the water is running.

Ask yourself this. If it is not does not specifically identified as being continious load, will it likely run for more then three hours? How will it be used? Also you want to keep the ambient temperture in mind as well