The main change would be that distance is considered in a different way.
"Same Island" is close.
"Same Group of Islands" is reasonably close.
"Everywhere else" is further away.
The extra work involved in getting a boat, etc means that people are naturally separated into tribes with defined geographical areas. This also means that there would be a long history of tribes raiding each other (for women, food, spoils, etc) but conquering each other would be harder to manage. One tribe would tend to hold one island and have a long tradition there.
As technology increases this would change, allowing people to start forming both empires and federations. The patterns would be similar to what we see on earth but with one big difference. The natural borders in islands mean that conquest tends to be all or nothing. You either claim an entire island or none of it. Situations like Cyprus which is split between two powers would be unusual.
The other big change is that the sea and water would have a lot of prominence in the culture. You should expect nautical terms and skills to have high social standing, with a skilled fisherman or diver held to high esteem. Ocean travel , navigation and fishing would all be incredibly important as it is the only source of commerce and the main source of food for the inhabitants.
I recommend looking at the culture of places like Hawaii, Japan, the Falkland Islands, and various other island chains throughout the world for inspiration.