Hello,
Theoretically you could describe a instruction set and make a processor using HDL language. But the behavior of this processor have some real-life limitations:
1) The target processor you described have to fit in the logic you have in the FPGA (FPGA does not have infinite logic and because it is programmable, with predefined cells, it will waste more logic than a fully customized ASIC)
2) The FPGA has speed limits (IOs, general logic, DSP, BRAM, etc.). Usually you can find these limits at target datasheet.
As the result of 1) and 2), if you implement a processor which fits an FPGA, the odds are you implement a much slower processor than a "newerish video game console", *if* you describe the processor flawlessly (which is a very difficult task).
Moreover, the processor is only one of several necessary parts of console. You need also some kind of ROM storage (one SSD, HDD, disk for older consoles, or even cartridges), and a volatile memory (SDRAM or some of several kinds of DDR{1/2/3/4} modules). Memories, in general, are prebuild in FPGA and divided in small BRAM modules, so you cannot have as much memory you want. Usually, you can have enough memory for cache in a FPGA and that is it.
So, once you cannot embed memory, you have to think at least on some kind of flash/ram controller + caches added to the processor. (+ interfaces with keyboard, control, power supply, etc.). All this put, the effort to develop this from scratch is *huge*.