You don't "buy a serial number" for Photoshop CC. You subscribe via your Adobe ID. No serial number is provided, though subscription can be handled via a redemption code. Ultimately the software has you "log in" with your Adobe ID then verifies your subscription online.
I have heard people say it can take some time for the subscription activation to occur after the transaction is complete. I believe when I redeemed my redemption code it worked right away, but that was a couple of years ago.
You do, of course, have the ability to download and run the software for 30 days for free, without subscribing to anything, so if you haven't completely used up your free trial period that gives you some flexibility.