Is this board Latin-style?
Jelly, 24 bytes

,ZµŒgaS$€F)Zṣ€0Ṣ&lt;JƊ€€)FẸ


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A monadic link taking as its argument the grid as a matrix with 1 representing squares needing a number and 0 representing the spacers. Returns 0 if the grid is Latin and 1 if not. 

Explanation

Main link, takes a matrix of 1s and 0s as its argument, z

,                         | Pair with:
 Z                        | - z transposed
  µ                       | Start a new monadic chain
                     )    | For each matrix (z and transpose(z)):
          )               | - For each row:
   Œg                     |   - Group runs of equal elements
       $€                 |   - For each group:
     a                    |     - And with:
      S                   |       - Sum of group
         F                |   - Flatten
           Z              | - Transpose
            ṣ€0           | - Split each at zeros
                  Ɗ€€     | - For each group:
               Ṣ&lt;         |   - Check whether sorted group is less than:
                 J        |     - Sequence along group
                      F   | Finally, flatten
                       Ẹ  | And check if any group returned true


Alternative in case true=true/false=false required:

Jelly, 24 bytes

,ZµŒgaS$€F)Zṣ€0Ṣ_JƊ€€)AƑ


Try it online!