Just having an open bitcoin server at domain.com that users could send unidentified payments to would be too much of a liability. Regular users aren't used to the idea of having to identify the payment. Merchants would get too many blank payments followed by "I paid you, where's my stuff?!" a week later.
The payment sequence does have a step where the receiver verifies the order before accepting it. It can reject the payment and return an error message if it doesn't contain a valid order number. That would require a difficult level of integration of custom code with the bitcoin server though.
It could be used the emule style and register a protocol handle for bitcoin client. So the shop could create link as such:
bitcoin://address-to-pay-to|amount|order hash or id|some other fields
In the end it would resume to such: