This may be caused by Having the serial number active for the install of Manga Studio and having a separate install of CSP may be confusing the license checker (what ever it's called). What I would do is to make backups of my Manga Studio files (don't forget to back up your custom brushes and materials!) and then delete Manga Studio from my system. Do the same for the Clip Studio Paint. Then reboot your machine. Install Clip Studio Paint and then see if the serial is accepted. If it is, great, if not, I'd contact SM Tech Support (or check the online knowledge base) for help. Or if you don't want to go through the hassle of deleting and then re-installing, just go to Tech Support and see if they can help you.
Be warned, though, that if you bought Manga Studio from e-bay or from any source not "blessed" by Smith Micro or Celsys, you may not be able to update to CSP as there was someone on the (unofficial) Manga Studio/Clip Studio Paint group on Facebook that had the same issues as you and was stuck because they bought their copy of MS from eBay.
Looks like your connection to Graphics Forum was lost, please wait while we try to reconnect.