Matrix Jigsaw Puzzles

APL (Dyalog Extended), 19 18 17 bytes

-2 thanks to ngn.

Anonymous tacit infix function. Takes n as left argument and list of two matrices as right argument. Requires zero-indexing (⎕IO←0). Incidentally, this function works on arrays of any number of dimensions.

≡.{∧,⍵⊂⍨⊂0=⍺|⍳≢⍵}

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≡.{…} identical results of the following function applied to each matrix ⍵ with n as ⍺?

 ≢⍵ size of matrix

 ⍳ indices 0…size–1

 ⍺| division remainder when divided by n

 ⊂ enclose to use along all dimensions

 ⍵⊂⍨ use that to partition* the matrix into a matrix of submatrices
  ^{* begins new partition when corresponding element is less than the previous; removes elements marked by zero}

 , ravel the matrix into a list of submatrices

 ∧ sort ascending

Python 2, 108 103 bytes

lambda s,*a:len(set(`sorted(sum([zip(*[iter(zip(*l))]*s)for l in zip(*[iter(m)]*s)],[]))`for m in a))<2

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Perl 6, 94 68 63 bytes

{[eqv] map *.rotor($^a).map({[Z] $_}).flat.rotor($a²).sort,@_}

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Anonymous code block that takes input as size, [matrix1, matrix2] and returns a boolean True/False. There might be a more efficient way of splitting the matrix into chunks than rotor.

Explanation:

{                                                            }  # Anonymous code block
       map                                                ,@_   # For both matrices 
           *.rotor($^a)   # Split the matrix into N sized chunks
                       .map({[Z] $_})  # Then zip each of those chunks together
                                     .flat  # Flatten the resulting list
                                          .rotor($a²)  # Then split into the NxN lists
                                                     .sort   # And sort them
 [eqv]    # And then check if the lists are equivalent