PHP configuration files are order dependent. If the same parameter settings exists in more than one file the last one read will take precedent. It sounds like the client's php.ini file is read first and your default is read second. This sounds odd, and Plesk is not something with which I am intimately familiar, but it would explain what you are seeing.
If your client has access to the source code they can use the New Relic PHP API method call newrelic_set_appname (name,license) to set a unique name as well as their own license key.