How can I tell if this appliance, a kiln, is using 208V single phase or three phase?
Shut off the breakers, and open the access panel on the kilns, and then examine what is connected to what, with great care, and a kiln manual in hand. Skutt has good manuals, in my experience.

Since they can be wired in many different ways, and may no longer match the "as shipped from factory" condition or the "as Sharpied by someone, sometime" condition, you have several options. As you are presently in a studio with 3-phase power available, unless you anticipate moving to a studio without three phase power available, it would be an ideal time to wire them as 3-phase when replacing the elements, if they are not presently so wired. If they currently connect to triplexed breakers in a 3-phase panel, odds are very good that they are already wired that way - but examining what is connect to what will tell the truth.

You could also check the labeling on the kiln-sitter and/or fancier computer control unit, if so equipped. The original factory voltage/phase configuration should be stamped into a steel plate on the kiln itself, as well.

Finally, if in doubt, hire a competent electrician or kiln technician (your ceramics supplier will either have one or know one) if you are not confident of getting it right yourself. Being sure is worth it on something this critical, and it sounds like you let "your old studio mate" handle this in the past without being directly involved yourself. Pay attention to what they do, and why, and document anything that needs documenting.

If the "time to element failure" is not quite a few years, also review the kiln operators manual and be sure that you are conforming to good operating procedures for best element/insulation lifetime. They should last quite a long time.