Conditional matches in sets of 3-tuples

If I understood you correctly, something like this would work

Cases[data,{{_,a_,b_},{a_,b_,c_},{b_,c_,_}}]

(* {{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
    {{0, 1, 0}, {1, 0, 1}, {0, 1, 1}}, 
    {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}}} *)

Select[Most[Rest /@ #] == Rest[Most /@ #] &] @ data

{{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}},
 {{0, 1, 0}, {1, 0, 1}, {0, 1, 1}}, 
 {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}}}

Pick[
 data,
 MapThread[SameQ, {
   data[[All, ;; -2, {2, 3}]],
   data[[All, 2 ;;, {1, 2}]]
   }]
 ]

(* Out: {{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, {{0, 1, 0}, {1, 0, 1}, {0, 1, 
   1}}, {{1, 0, 1}, {0, 1, 1}, {1, 1, 1}}} *)