If you purchased it from Google Play, it should be available immediately for download. Apps are registered to the Google account they were purchased under; no real serial that I know of. (At least that's how several of my purchases were done in Google Play, including PS Touch.)
Warunicorn, Thanks for the reply. It still leaves me in a position where I can't register the product with Adobe. The reason this is an issue with me is that years ago I bought, at retail, a version of PhotoShop (Vers. 6) and didn't register it at the time. Since then I have been unable to update, upgrade or take advantage of Adobe support. Of course any documentation and media from the original purchase (around 2001) have gone missing so I try to 'follow the rules' with new purchases.
Technically, your app is registered under your Google account. From there, it's up to Adobe to either update the current app or release a new version. (I've only seen a couple instances where new versions of apps were released separately.) PS Touch has adhered to the former, so far. (And it's actually come a ways since its initial release both in form and function.)
Photoshop serial numbers for the desktop are not compatible with Photoshop Touch if this is what you are trying to figure out.
There is no registering of the application in the same sense. If you bought it on the Google Play Store and have that account on your device
then you should be able to use it once installed.
If you really want to register Photoshop version 6 (i.e. 7 years old version of Photoshop) you should probably want to try out the Photoshop forum.