Can't rename Azure App Service plan using the Azure Portal

No, you can't.

However, if you create a new App Service Plan in the same region and resource group then you can move all your sites to the new App Service Plan and delete the old one.

No but you can change it like this (if it is in the same region and resource group as mentioned by Zain Rizvi):

Create a new App Service Plan and bind it to the correct Subscription and Resource Group. Example:

Then go to your App Service and choose "Change App Service plan"