How can I allow access to two user accounts using one login?

I would reverse engineer the User Switching plugin and then create a solution that works for you. Fortunately there is not a whole lot of work to do. Most of the crucial functionality is in this function:

function switch_to_user( $user_id, $remember = false, $set_old_user = true ) {
    if ( ! $user = get_userdata( $user_id ) ) {
        return false;
    }

    $old_user_id = get_current_user_id();

    if ( $set_old_user && $old_user_id ) {
        user_switching_set_olduser_cookie( $old_user_id );
    } else {
        user_switching_clear_olduser_cookie( false );
    }

    wp_clear_auth_cookie();
    wp_set_auth_cookie( $user_id, $remember );
    wp_set_current_user( $user_id );

    if ( $set_old_user ) {
        do_action( 'switch_to_user', $user_id, $old_user_id );
    } else {
        do_action( 'switch_back_user', $user_id, $old_user_id );
    }

    return $user;
}

You could just maintain a list of “linked accounts” for each user to determine if a user is allowed to switch to another account.

Pseudo code:

$switch_to = 0; // some user id
$linked_accounts = get_linked_accounts( get_current_user_id() );
if(in_array($switch_to, $linked_accounts)) {
    switch_to_user($switch_to)
}

You would need to define get_linked_accounts and setup a way to actually do the linking. You should also do a lot of testing to make sure users will not be able to switch to any arbitrary account.