Lets define the term:
Firmware - In electronic systems and computing, firmware is a type of computer program that provides the low-level program control for the device's specific hardware. Firmware can either provide a standardized operating environment for the device's more complex software, or, for less complex devices, act as the device's complete operating system, performing all control, monitoring and data manipulation functions. Typical examples of devices containing firmware are embedded systems, consumer appliances, computers, computer peripherals, and others. Almost all electronic devices beyond the simplest contain some firmware.
So you could think it would be better suited to call the coding of an FPGA 'Firmware' but the coding is not permanent unless the code is stored in external eeprom. It is more of programmable hardware hence the words in FPGA Field Programmable indicates that it is software controlled in some fashion of Gate Array or could be seen as hardware.
When you look at PGAs they almost follow the path of ROMs, where PGAs in the beginning were received from the factory by the programmer to burn in the hardware configuration, a similar procedure for PROMs then EPROMs and later EEPROMs which later was called firmware as the code aka software would stay permanent when power is off.
So is it called 'Firmware' for FPGAs, i think not as it is not permanent, however, PGAs once burned they are permanent.
You could view FPGAs as something similar to RAM. The FPGA powers up and goes through a startup routine, loads up a configuration from a storage medium, rearranges the hardware according to its instructions and loses its setup when power is turned off or reset.
So i think it lies in between Software, Firmware and Hardware maybe a collection of the three call it 'SoFiHa' a new term for FPGAs.