I'm always afraid in case of list-manipulation that there was a duplicate in the past. But I do not remember.

You can try this:

Extract[list, Position[labels, #]] & /@ Union@labels

{{{1 ,2}, {-3, 4}, {-9, 1}}, {{5, 6}, {7, 8}}, {{-1, 3}, {0, 1}}}

variation

GatherBy[Sort@Thread[Rule[labels, list]], First][[ ;; , ;; , 2]]

{{{-9, 1}, {-3, 4}, {1, 2}}, {{5, 6}, {7, 8}}, {{-1, 3}, {0, 1}}}

I'm always afraid in case of list-manipulation that there was a duplicate in the past. But I do not remember.

You can try this:

Extract[list, Position[labels, #]] & /@ Union@labels

{{{1 ,2}, {-3, 4}, {-9, 1}}, {{5, 6}, {7, 8}}, {{-1, 3}, {0, 1}}}

and this:

Pick[list, labels, #] & /@ Union@labels

{{{1, 2}, {-3, 4}, {-9, 1}}, {{5, 6}, {7, 8}}, {{-1, 3}, {0, 1}}}

GatherBy variation

GatherBy[Sort@Thread[Rule[labels, list]], First][[ ;; , ;; , 2]]

{{{-9, 1}, {-3, 4}, {1, 2}}, {{5, 6}, {7, 8}}, {{-1, 3}, {0, 1}}}