I got the result using array_intersect

$x = Array
(
  0 => 'sku',
  1 => 'qty'
);
$y = Array
(
  0 => 'sku',
  1 => 'qty'
);

$z = array_intersect($x,$y);

print_r($z);

Output will be

Array
(
[0] => sku
[1] => qty
)

Just check the content of array_intersect($array1, $array2)

$result = count(array_intersect($array1, $array2)) > 0;

If you want to get 'nature' first, because 'walk' comes before 'horse', you need to iterate over the words first, not over the tags.

$tags = [
    'animals' => ['cat', 'dog', 'horse', 'ferret'],
    'nature' => ['walk', 'outdoor', 'tree', 'plant'],
];
    
    
function getTags(string $text, array $tags): array
{
    $lowerC = strtolower($text);
    $str = preg_replace("/[^A-Za-z\'\- ]/", '', $lowerC);
    $arrayT = explode(" ", $str);

    $tagArray = [];
    foreach ($arrayT as $word) {
        // find tag for this word
        foreach ($tags as $cat => $values) {
            if (in_array($word, $values)) {
                // append the tag to the list
                $tagArray[] = $cat;
            }
        }
    }
    
    // remove duplicates
    return array_unique($tagArray);
  
}


$res = getTags('During my walk, I met a white horse', $tags);
var_dump($res);

Output :

array(2) {
  [0]=>
  string(6) "nature"
  [1]=>
  string(7) "animals"
}

EDIT

As @GeorgeGarchagudashvili mentionned, the code could be optimized by preparing an array for comparison. Here is a way :

function getTags(string $text, array $tags): array
{
    $lowerC = strtolower($text);
    $str = preg_replace("/[^A-Za-z\'\- ]/", '', $lowerC);
    $arrayT = explode(" ", $str);
    
    // Prepare tags for searching
    $searchTags = [];
    foreach ($tags as $cat => $values) {
        foreach ($values as $word) {
            $searchTags[$word] = $cat;
        }
    }

    $tagArray = [];
    foreach ($arrayT as $word) 
    {
        // find tag for this word
        if (isset($searchTags[$word])) 
        {
            // append the tag to the list
            $tagArray[] = $searchTags[$word];
        }
    }
    
    // remove duplicates
    return array_unique($tagArray);
  
}

You have no duplicate values in your array. From the manual page:

Note: Two elements are considered equal if and only if (string) $elem1 === (string)$elem2 i.e. when the string representation is the same, the first element will be used.

However, there is a solution in the user-contributed notes on that page:

function array_iunique($array) { 
    $lowered = array_map('strtolower', $array); 
    return array_intersect_key($array, array_unique($lowered)); 
}