create multidimensional array using a foreach loop

You're aaalmost there. Just add the level to the array creation :)

$levels = array('low', 'medium', 'high');
$attributes = array('fat', 'quantity', 'ratio', 'label');

foreach ($levels as $key => $level):
       foreach ($attributes as $k =>$attribute):
             $variables[$level][] = $attribute . '_' . $level; // changed $variables[] to $variables[$level][]
       endforeach;
endforeach;

echo '<pre>' . print_r($levels,1) . '</pre>';   
echo '<pre>' . print_r($variables,1) . '</pre>';  

Output

Array
(
    [low] => Array
        (
            [0] => fat_low
            [1] => quantity_low
            [2] => ratio_low
            [3] => label_low
        )

    [medium] => Array
        (
            [0] => fat_medium
            [1] => quantity_medium
            [2] => ratio_medium
            [3] => label_medium
        )

    [high] => Array
        (
            [0] => fat_high
            [1] => quantity_high
            [2] => ratio_high
            [3] => label_high
        )

)

$levels = ['low', 'medium', 'high'];
$attributes = ['fat', 'quantity', 'ratio', 'label'];

$result = [];
foreach ($levels as $level) {
    $result[$level] = [];
    foreach ($attributes as $attribute) {
        $result[$level][] = $attribute . '_' . $level;
    }
}

var_dump($result);