Can it run software like a real Amiga or Atari / C64 or Spectrum etc?

Yes, the software that is loaded onto the replay comes in whats known as cores. eg: an OS (operating system/ loader) if you like is stored into a file which might be (amiga OS)  amiga_replay.bin or amiga_replay.bit and when the replay boots on the sdcard, if you were using an amiga core, it loads the kickstart as an amiga did and waits for you to either load a floppydisk or hdisk image. The cores are if you like reverse engineered in a fashion and then converted onto files as as to make them as close as possible to the original machines as can be. Theoretically you should be able to load most games and applications for your operating system or core desired and it should run, but because of compressing routines and other machine hacks, at this time some still may not work. The cores are a work in progress and get updated from time to time. Always check back to make sure you are using the latest versions.