"Movable using casters" instantly qualifies it as furniture, and not part of the building's structure. Heck, if it were a rental unit, you could take it with you.
As for electrical to this rolling cabinet, that's a trick. It's extremely likely that a floor-wiring-based solution is going to get crunched and mashed. You would have to install a floor outlet consistent with NEC of course.
But for guidance on the cords to the island, that's outside NEC's bailiwick and falls under UL. There's no enforcement/approval mechnanism for portable furniture, but on the other hand, there's no certification mechanism either. That basically means if the fire inspector is going through the ashes and finds the house fire started inside the electrical on your homebuilt portable furniture, that's between you and your fire-insurance company.
Consider pendants to connect power. Pendants are cords that hang from the ceiling, and have competent, industrial grade strain relief on both ends. For instance, you could fit aluminum junction boxes hanging down low, and plug your appliances directly into those. Remember kitchen appliance cords are only 2' long. Or such a pendant might have an L14-20 twist-lock connector, and then go into a multi-wire branch circuit in your furniture.
If you were exceptionally clever, you could have the pendant tie into the ceiling at any of several L5-20 or L14-20 twist-lock connectors, and move the pendant around as you move the table around. I for one would hide those connectors by putting spotlights right next to them (anyone trying to look at them would get an eyeful of light :)
As for wiring inside furniture, I generally do that one of two ways.
- I simply glue COTS power strips to the furniture, which is factory approved since the power strips have mounting slots for this purpose.
- I wire entirely in the EMT metal conduit wiring method, securely anchored to the furniture. I then either use very large cordage coming properly off a knockout in a junction box, with the best strain relief the electrical supply house (never Home Depot) has to sell me, or I fit an inlet and use a COTS heavy extension cord. The inlet has the demerit of falling out easily.