you are correct. if you were properly signed in to your cc desktop app (using the same adobe id used to subscribe to cc) and your computer was able to connect to adobe, you would not be prompted for a serial number.
so, sign out and then back in to your cc app, Sign out, Sign in | Creative Cloud Desktop app
if that fails, Sign in, activation, or connection errors | CS5.5 and later
p.s. that's not a mistake. don't uninstall anything for a few months. if you then find you have no need for the older ps cc, you can uninstall it.