Syntax

The issue with your WordPress approach is the get_var calls used to get the subscriber id and interest name. get_var is intended to get just a variable from the database like the number of rows in the table or sum of a numeric column in the table, see the WordPress Codex

What your looking to do is select a row, using get_row

public function update_subscriber_interest($subscriber_id, $interest_name) {
global $wpdb;

$select_subscriber_id = $subscriber_id;
$select_interest_name = $interest_name;

$get_subscriber_id = $wpdb->get_row(
    "SELECT * FROM {$wpdb->prefix}subscriber WHERE id =   ".$select_subscriber_id.""
);

$get_interest_name = $wpdb->get_row(
    "SELECT * FROM {$wpdb->prefix}interest WHERE name="".$select_interest_name."""
);

$sql = $wpdb->insert(
    "{$wpdb->prefix}subscriber_interest",
    array(
        "id"    => $get_subscriber_id->id,
        "name"  => $get_interest_name->name
    ),
    array(
        '%d',
        '%s'
    )
);
return $sql;

}

Table Structure

The table structure you describe is going to give you an subscriber_interest table that links to the interest table using a non-unique column the name, so if there are two seperate interest rows that are both named RSVP things are going haywire.

I’d recommend using the id column to link the interest table to the subscriber interest.