There is nothing technologically hard in issuing certificates. It is just a bit of encoding, and a signature; a basic PC can do that a thousand times per second. While CA don't normally indulge in on-demand certificate issuing, there is no inescapable technical barrier to it.
The most plausible explanation is that you are currently connecting through a proxy which does a man-in-the-middle "attack" and really generates a fake certificate on demand. Not a real attack in the sense of some fraudster trying to pick on you; more an "institutional" attack by the local sysadmin, running a product like that one. This is becoming popular in corporative environments.
To check whether this occurs, a simple trick is to change your system date. Indeed, one conundrum of MitM proxies is that they must generate a certificate which will be accepted by the client, which, in particular, implies that the certificate must be such that the current time, as believed to be by the client, falls within the validity range. An often overlooked property of SSL is that both client and server tell to each other their own notion of the current time, in the early stages of the handshake procedure. This allows the server to know how the client's clock is set. If you change your current date and time to, say, last week, and the server's certificate follows that change (close all browser windows and restart it to ensure that a new SSL session is used), then the server's certificate was definitely produced on demand, because there is no other way that certificate would adjust to a wildly off client clock.
You will also want to see to what trust anchor (aka "root certificate") the server chain goes up. If an institutional MitM is in force on your system, then the chain will go up to a corresponding "rogue" root CA. What should happen with GoDaddy's certificates is described here (of course, the MitM may theoretically alter that page as well, as you see it; but if there is a discrepancy between what GoDaddy states and what you observe then it is proven that there is some ongoing foul play).