Instead of an FTDI FT232, the Nano Every uses a SAMD11D14A microcontroller for its USB interface. So you are correct that you can't use the FT_Prog application to change the name.
The name is defined in the firmware that runs on the SAMD11 chip. This was nicknamed "MuxTO" by Arduino. The cool thing is the MuxTO firmware is an Arduino sketch:
which means you can compile and upload it to the SAMD11D14A using the Arduino IDE, as described here:
For this purpose, you need a hardware package that provides SAMD11 support to the Arduino IDE. There is a nice 3rd party hardware package for this created by mattairtech and Arduino maintains a fork of it in the muxto branch of the Arduino SAMD Boards repository:
So once you have that installed, there is a Tools > Board > Arduino MuxTO option in the Arduino IDE. I believe the USB name of the Nano Every is defined in the boards.txt file of that hardware package in the muxto.build.usb_product property:
muxto.build.usb_product="Arduino Nano Every"
So if you modify that line and then upload the MuxTO firmware to the SAMD11 on your Nano Every, I think the name will be changed accordingly. Note that you need to restart the Arduino IDE before changes to the boards.txt file will be recognized.