Find duplicates in list of InfiniteLine

DeleteDuplicates[lines, MemberQ[{##},RegionIntersection @ ##]&]

{InfiniteLine[{{0, 0}, {1, 0}}], InfiniteLine[{{0, 1}, {1, 0}}]}

Also

DeleteDuplicatesBy[VectorAngle @@ #[[1]] &]@lines

{InfiniteLine[{{0, 0}, {1, 0}}], InfiniteLine[{{0, 1}, {1, 0}}]}

Here's another approach:

DeleteDuplicates[lines, And @@ RegionMember[#, #2[[1]]] &]

(* {InfiniteLine[{{0, 0}, {1, 0}}], InfiniteLine[{{0, 1}, {1, 0}}]} *)

It still feels a little bit wasteful for me to use a trigonometric function so there's room for improvement, but it's not as wasteful as bringing to bear region functionality on this problem:

sameLine[InfiniteLine[{u1_, u2_}], InfiniteLine[{v1_, v2_}]] := With[{u = u1 - u2, v = v1 - v2},
  PossibleZeroQ[VectorAngle[u, v]] || PossibleZeroQ[VectorAngle[u, v] - Pi]
  ]

DeleteDuplicates[lines, sameLine]

{InfiniteLine[{{0, 0}, {1, 0}}], InfiniteLine[{{0, 1}, {1, 0}}]}\

Here is a version with just the square root and multiplication:

sameLine[InfiniteLine[{u1_, u2_}], InfiniteLine[{v1_, v2_}]] := With[{u = u1 - u2, v = v1 - v2},
  PossibleZeroQ[Abs[Dot[u1 - u2, v1 - v2]/(Norm[u1 - u2] Norm[v1 - v2])] - 1]
  ]

or another even simpler form:

sameLine[InfiniteLine[{u1_, u2_}], InfiniteLine[{v1_, v2_}]] := 
 PossibleZeroQ[Dot[u1 - u2, {-1, 1} Reverse[v1 - v2]]]

All of the previous functions have the flaw that they don't count parallel lines as duplicates. We can fix that by adding another condition:

sameLine[InfiniteLine[{u1_, u2_}], InfiniteLine[{v1_, v2_}]] := And[
  PossibleZeroQ[Dot[u1 - u2, {-1, 1} Reverse[v1 - v2]]],
  PossibleZeroQ[Dot[u1 - v1, {-1, 1} Reverse[v1 - v2]]]
  ]