Jelly, 9 bytes

S;§>Ƈ1S=L

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Uses Unrelated string's test suite.

Explanation

S;§>Ƈ1S=L
S         sums of columns
 ;        concat with
  §       sums of rows
    Ƈ     keep the sums which are:
   > 1     greater than 1
      S   sum of that
       =L = input length?

Jelly, 10 bytes

S;§>Ƈ1⁼LW$

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Uses Unrelated string's test suite.

Fixed with some help from Unrelated string.

Explanation

S;§>Ƈ1⁼LW$
S           sums of columns
 ;          concat with
  §         sums of rows
    Ƈ       keep the sums which are:
   > 1       greater than 1
      ⁼LW$  = [length] ?