I have tried on Windows, Ubuntu and CentOS by running the installer without stopping the Splunk UF process and it seems the installer will stop the process and then proceeds with replacing the files and upgrade steps. I havent seen any errors post upgrading on the logs.
That could be. On windows I know that is indeed the case, on linux there are various installer options, while many people also just manually unpack and deploy the tgz archive. I guess that is why the docs choose the better safe than sorry option and recommend to stop splunk before installing, so there is no dependency on the particular installer being used.
Also, it is good to realize that even if the installer does it for you, the install does involve stopping and starting splunk.