Negative meta_query if storing multiple post_meta values with shared meta_key

Thats because you created the meta_query property incorrectly, it should be a multi diminsional array.

$pet_owners = new WP_Query([
    'post_type' => 'pet_owner',
    'posts_per_page' => -1,
    'meta_query' => [
        [
            'key' => 'pet',
            'value' => 'cat',
            'compare' => '!='
        ]
    ]
]);

For more information about meta_query see Custom Field (post meta) Parameters