From the rules :
Catan Seafarers Game Rules and Scenarios
If you build (or move) a ship on an edge next to a harbor, you take
the harbor. If you have a settlement on the coast, you must
immediately place the harbor on an adjacent edge. However, harbors
must never occupy adjacent or the same edges. In case you have no
appropriate coastal settlement, you can put the harbor aside until
such a settlement is built. When you have placed a harbor, you can use
it immediately, even during the same turn.
Interpretation. This means that the first player who places or builds a ship next to a harbor, physically picks up the harbor, i.e. removes it from the board. The player will then place this harbor on an edge next to one of his/her coastal settlements on the main island, if the player has one. No one can build a settlement on any of the small islands.
«So, in this case who will have the final claim and use of the harbor on the new island?»
The player who moved next to the harbor claims it (picks it up) and gets to place the harbor next to one of his/her coastal settlements, immediately, or later.
«But, we have another player who is gonna place a settlement for first time on the same island.»
During play, no settlement can be built on the surrounding small
islands that do not produce resources.
For the harbor, it's first come, first served. And, no player can build a settlement on any of the small islands.