Inserting a name into a database table
In terms of security, you should be safe from SQL injection since you are using parameterized queries as recommended.  That's true whether or not you validate the names using the regex.  Do not perform any additional escaping — that would only mangle your data.

That regex is for enforcing your business rules (i.e. you want to reject names written in Cyrillic, names with French accents like é, Irish surnames like O'Something), and has nothing to do with database security.

I do not recommend mixing PDO with the WordPress database API.

The WordPress documentation recommends that you use $wpdb-&gt;insert() for simple INSERT queries.

In accordance with the WordPress documentation, you should check the return value from $wpdb-&gt;query() — a FALSE value indicates failure.  You should do that before declaring victory with echo json_encode(['status' =&gt; true]);.

if (!empty($errors)) {
    echo json_encode(['status' =&gt; false, 'errors' =&gt; $errors]);
} elsif (FALSE === $wpdb-&gt;insert('people', ['name' =&gt; $_POST['name']], '%s')) {
    echo json_encode(['status' =&gt; false, 'errors' =&gt; ['Database error: ' . $wpdb-&gt;last_error]]);
} else {
    echo json_encode(['status' =&gt; true]);
}