In short, no, one circuit for the all the items in your Island will suffice and the final answer is in your actual local Electrical codes, but as rule of thumb, one will work for all. You may be required a GFI circuit, which can be a pain but they do protect agains potential shock.
There are some exceptions, for example, you may have a coffee maker, and or a warming drawer in your island. You do need to consider the potential power or current load factors, which can be easily figured by adding up the current and or wattage items use.
Most island Refers draw less than 3amps running, and 5-6amps at starting, and your smallest wired circuit is typically 15amps, so you have room for other items without exceeding the circuits capacity.
I.E. a warming drawer and or a coffee maker, can be about 8-10amps or 800-12000 watts, so you may need two circuits if you have a warming drawer. This is simple calculations for any electrician or experienced handyman.