How to select 2 pairs of distinct twins form a list of 4-tuples?

Another way

Select[data, Values[Counts[#]] === {2, 2} &]

Try this:

Pick[data, Values[Counts[#]] === {2, 2} & /@ data]

A pattern based approach:

Cases[
 data,
 {x_, x_, y_, y_} | {x_, y_, x_, y_} | {x_, y_, y_, x_} /; x != y
 ]

Select[
 data,
 MatchQ[Sort[#], {x_, x_, y_, y_} /; x != y] &
 ]

Another approach is to construct the desired tuples directly without selection from a larger set:

Subsets[Range@6, {2}] // 
 Map[Join[#, #] &] // 
 Map[Permutations] // 
 Flatten[#, 1] &

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