For a kitchen island receptacle circuit, the breaker has to match the wire size and the way the circuit is being used, not just the fact that it is in an island. In most modern kitchen layouts, island outlets are served by a 20 amp branch circuit because kitchen receptacle circuits are typically required to handle countertop-style loads like mixers, toasters, and other small appliances. If the circuit is wired with 12-gauge copper, a 20 amp breaker is the usual match. If it is wired with 14-gauge copper, that would normally be limited to a 15 amp breaker.
The key point is that the breaker protects the wire, not the receptacle itself. A 20 amp breaker on 14-gauge wire is the wrong combination because the wire could overheat before the breaker trips. On the other hand, putting a 15 amp breaker on 12-gauge wire is electrically safe, but it may be unnecessary if the circuit is intended to serve typical kitchen appliance loads. For a kitchen island, many electricians lean toward 20 amp because the load can be heavy even if there is only one or two receptacles.
Another thing that matters is whether the island outlet is on its own dedicated circuit or shared with other kitchen receptacles. If it is sharing a circuit with other general-use countertop receptacles, you need to be careful not to overload it, especially with coffee makers, air fryers, and microwaves all pulling power at the same time. In practice, a dedicated 20 amp circuit for the island is often the cleaner choice.
Also check whether the outlets need GFCI protection, which is commonly required in kitchen areas, including islands. Depending on the setup, that protection can come from a GFCI breaker or a GFCI receptacle upstream. That does not change the breaker amp rating, but it does affect how the circuit is built.
So the short answer is: most kitchen island outlet circuits use a 20 amp breaker with 12-gauge copper wire. Use 15 amp only if the circuit is wired with 14-gauge copper and the installation is allowed to be that size. If you are unsure about the exact requirements in your area, it is worth checking the local electrical code before closing everything up. If you want, I can also help you figure out the right breaker based on the wire size and whether the island has one or two receptacles.