I created a page and inserted a database.

When I insert to right column some widgets, like list of categories, best records, etc, I see the same widgets on every section, doesn't matter if I see database index, or category, or record...

Is there way how I could setup widgets for example on index I would see only widget about best recordw, when viewing record I would see only list od categories, etc... Is it possible?

I understand it's basically still the same page.

It's unfortunately not possible out of the box yet, but there's at least one 3rd party application in the marketplace, which sounds like what you described.