Query a meta key using an array of values where the database value is a string

You could use multiple meta clauses in your WP_User_Query, along with a LIKE comparison.

Something like

$args = array(
    'role'  => 'member',
    'meta_query' => array(
        'relation' => 'OR',
        array(
            'key'     => 'specializations',
            'value'   => 'doctor',
            'compare' => 'LIKE',
        ),
        array(
            'key'     => 'specializations',
            'value'   => 'researcher',
            'compare' => 'LIKE',
        ),
    ),
);
$found_users = new WP_User_Query($args);

In reality you’d probably want to loop through your target specializations and build the set of clauses dynamically.

That said, be aware that queries using LIKE are expensive. I’d say this method is a last resort, assuming you cannot refactor the DB structure to store each specialization in its own row (serialization won’t help you here), and assuming that there are too many records for you to just grab the lot and search them in your PHP.