Dropping list elements from nested list after evaluation

If you prefer using DeleteCases,

list2 = DeleteCases[list1, _?(Abs[#[[3]]] == 1 &)]

{{1, 1, x, 2 x, 3 x}, {1, 1, -x, -2 x, -3 x}}

Delete[
 list1, 
 Position[Abs[list1[[All, 3]]], 1]
 ]

{{1, 1, x, 2 x, 3 x}, {1, 1, -x, -2 x, -3 x}}

if you want to use Select, try this

Select[list1,!NumberQ@#[[3]]||Abs[#[[3]]]!=1&]

{{1, 1, x, 2 x, 3 x}, {1, 1, -x, -2 x, -3 x}}

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