Congratulations! You have discovered one of the most complained-about "features" of the Samsung Galaxy S family, which has spawned multiple threads on multiple forums.
The problem is that Samsung built its own Contacts app and installed it in lieu of the stock Android Contacts app, and in the process destroyed part of the Android/Google functionality. You are correct, this app does not display the groups assigned to contacts within Gmail. (I believe the group assignment is downloaded, but the app does nothing with it.) The "groups" that Samsung gives you have nothing to do with Google's Gmail contact groups.
You cannot uninstall the brain-dead Samsung app, but you can load a third-party app to use instead. Try Contacts Evolved, free in the Market. It is a little buggy, but far better in this regard than the fundamentally broken Samsung app. In Contacts Evolved, the contact groups are segregated into separate tabs.