Is my kids' alphabet mat properly grouped by colors?
MATL, 16 15 bytes

1e"G@=4&amp;1ZI1&gt;vzg


Input is a 2D char array (with rows separated by ;). Output is 0 if input qualifies, or 1 otherwise.

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Explanation

The code essentialy checks if each char in the input has only one connected component, considering 4-connectivity (that is, no diagonals).

Repeated chars are processed repeatedly (which is golfier than deduplicating).

1e       % Implicit input. Reshape into a row vector of chars
"        % For each char
  G      %   Push input again
  @      %   Push current char
  =      %   Equal (element-wise)? Gives a matrix of zeros and ones, where one
         %   represents the presence of the current char
  4      %   Push 4. This will indicate 4-connectivity
  &amp;1ZI   %   Matrix with labels of connected componnents. Inputs are a number (4)
         %   to indicate connectivity, and a binary matrix. The output is a matrix
         %   the same size as the input where each connected componnent of ones
         %   in the input is replaced by a different integer starting at 1
  1&gt;     %   Greater than 1 (element-wise)? The result is a matrix. If the result 
         %   is true for some entry the input doesn't qualify
  v      %   Concatenate vertically with results from previous iterations
  z      %   Number of nonzero/true values
  g      %   Logical. Converts nonzero to true
         % Implicit end. Implicit display. False / true are displayed as 0 / 1