Share or sync user table data with another user table

Welcome to WPSE. Asking for plugin recommendations is considered off-topic here, so here’s an action approach to the problem.

wp_insert_user() takes care of adding new users to the database. The last action the function fires is user_register, which “Fires immediately after a new user is registered”. You could hook your function to this action and use it to update the custom DB table.

In your action function you can use the global $wpdb to manipulate your database tables whether they are standard WP tables or not. (You can also use wpdb class to connect to other databases, too.)

To insert new data into the database you can use the wpdb::insert( string $table, array $data, array|string $format = null ) method.

If there’s some user data that needs to be kept updated and synced, then you can hook to profile_update action, which wp_insert_user() also fires, but “…immediately after an existing user is updated”.