You specified fields for both joins:
->join(array('up1'=>'user_profile'), "up1.user_id = request.request_user_id AND up1.key = 'user_first_name'", array('up1.value'), 'left')
->join(array('up2'=>'user_profile'), "up2.user_id = request.request_user_id AND up2.key = 'user_last_name'", array('up2.value'), 'left')
When query is converted to actual sql, these fields will be automatically namespaced, so you will get "up1.up1.value" in fields list.
Remove fields references from joins and it should work.
UPD. Right, there's more to it. You can't pass 'user_first_name' as a string value to the "on" parameter of join as it will be interpreted as a column name. So you have to pass an expression:
$select = $sql->select();
$select->from('request')
->columns(array('*', new Expression('CONCAT(up1.value,"#",up2.value) as display_name')));
$expressionString = '? = ? AND ? = ?';
$types = array(Expression::TYPE_IDENTIFIER, Expression::TYPE_IDENTIFIER, Expression::TYPE_IDENTIFIER, Expression::TYPE_VALUE);
$parameters = array('request.user_id', 'up1.user_id', 'up1.key', 'first_name');
$expression1 = new Expression($expressionString, $parameters, $types);
$parameters = array('request.user_id', 'up2.user_id', 'up2.key', 'last_name');
$expression2 = new Expression($expressionString, $parameters, $types);
$select->join(array('up1'=>'user_profile'), $expression1, array('value'), 'left')
->join(array('up2'=>'user_profile'), $expression2, array('value'), 'left');