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The characteristic that made the legs of Roman tables frequently adjustable was their capability to allow for height adjustments. This feature was significant in the context of Roman furniture design, as it provided flexibility for different uses and settings. The ability to modify the height ensured that the table could comfortably accommodate various activities, such as dining or working, catering to the needs of both the elite and the general populace.

In the design of Roman furniture, practicality was paramount. Adjustable legs made it easier to store tables or to align them with other pieces of furniture or the users' needs. The adaptability of the table design reflects the Roman emphasis on functionality, allowing for a range of uses in homes and public spaces.

The other characteristics listed, such as removable legs, interchangeability, and foldability, do not directly convey the idea of height adjustment, focusing instead on different aspects of portability and versatility. While these features may have their own merits, they do not pertain to the specific functionality of altering the table's height, which is the core focus of this question.