You can do it, but you cannot change the PID and VID without a new certificate.
Did you changed the name with the FT-Prog Tool (MProg is outdated)?
You can download the last version from here:
You can set everything on your FTDI chip, even change the PID and VID. But at this point, you will not be able to use the standard FTDI driver anymore as you need to sign it with a new certificate (it is about 400$/year).
What you can do is leave the PID and VID and just give another name. So you will be able to use the default driver and you will see your custom name in the device manager.
Of course, in the list you just see "USB Serial Port (COM X)". But in the description of the device (Serial port), you will find this string:
If you create an application, you can just check for this, to see if it is your module. If you want more, you have to develop and release your own driver. But you can't do it anymore without a certificate, or without start Windows in unprotected mode.