JavaScript (ES6), 60 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some(p=>m.some(q=>p!=q&&p.reduce((t,v,x)=>t+v*q[x],0)))

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JavaScript (ES6), 60 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some(p=>m.some(q=>p!=q&&p.reduce((t,v,x)=>t+v*q[x],0)))

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Commented

m => // m[] = input matrix
m.some(p => // for each row p[] in m[]:
 m.some(q => // for each row q[] in m[]:
 p != q && // test whether p[] is not equal to q[]
 // (comparison of pointers)
 p.reduce( // for each value v at position x in p[],
 (t, v, x) => // using t as the accumulator:
 t + v * q[x], // add v * q[x] to the accumulator,
 // which gives 1 if v = q[x] or -1 otherwise
 0 // start with t = 0
 ) // end of reduce() -> gives 0 if and only if
 // exactly half of the values are matching
 ) // end of inner some()
) // end of outer some()

JavaScript (ES6), 62 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some((p,i)=>m.some(q=>i--*p.reduce((t,v,x)=>t+v*q[x],0)))

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JavaScript (ES6), 60 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some(p=>m.some(q=>p!=q&&p.reduce((t,v,x)=>t+v*q[x],0)))

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JavaScript (ES6), 63 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some((p,i)=>m.some(q=>i--&&p.reduce((t,v,x)=>t+v*q[x],0)))

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JavaScript (ES6), 62 bytes

Returns false for Hadamard or true for non-Hadamard.

m=>m.some((p,i)=>m.some(q=>i--*p.reduce((t,v,x)=>t+v*q[x],0)))

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