The 680X0 CPU in your Amiga, as well as the various ECS/AGA chips, are silicon chips which are fixed in a permanent state. They were custom chips, designed for a specific purpose.
The FPGAs of today are like 'blank chips'. There's nothing at all in them. What you do, is at boot-time, you 'select' which 'soft core' you want 'loaded' into that FPGA chip. It (the FPGA chip) then becomes whatever you need it to become. In the case of Minimig and FPGA Replay etc, the 'soft core' contains the instructions necessary to turn that FPGA into a 68K CPU, complete with ECS, AGA chips etc.
So it goes from being a machine with a blank chip, to being a machine with a 680X0 CPU and AGA! And this is NOT emulation, it is done in real silicon, thanks to the FPGA chip! So now, when you run Amiga software, all it sees is the 68K CPU, the AGA chips, etc. and they will run no differently than if it was a 'real' Amiga, because it IS a 'real Amiga'. The FPGA reacts to Amiga software in much the same way as the 68K, Lisa, etc. chips in an A1200 would do.
With the FPGA Arcade Replay, you'll eventually have a choice of hardware platforms to choose from at boot-up: Amiga, Atari ST (gasp!), Commodore 64, Arcade machines, etc. The FPGA 'becomes' what you need it to become, and hey presto! You have a custom computer system created at boot time

The beauty of the FPGA is, however, that unlike 'real' chips, the FPGA code can be enhanced and improved, added to. AGA can become SuperAGA. You can eventually have a CPU with the compatibility of an 68020 but running software faster than a real 68040 or 68060 could, etc.
I hope this clears things up for you?