Given the way the board is reset by pressing the two top targets, it looks like it is using illuminated push buttons rather than anything more clever.
The hard part might be finding or making illuminated buttons which are both sensitive and robust enough.
Beyond that it's just a matter of choosing and lighting a random a LED, wait for the corresponding button to be pushed, then on to the next. That 'randomness' might have to be slightly more than simply 'random' or raw 'pseudo random'.
The harder part is determining how long it takes between lighting the LED and its button being pushed. But it should be possible to run an internal timer at microsecond rates while polling for a button push and keeping the display updated.
The differing games and variances can all be built on top of that base.
It all seems to be possible to me using any PICAXE with enough I/O. The LED's can be implemented with a short RGB LED strip and just two signal wires, the seven buttons would probably be best handled as a port wide input for speed of detection. A three-wire 7-segment display module is all that's needed for the output display.
So a 14M2, 20M2 or 20X2 would seem suitable to me. The 20X2 would make timing easier to handle and, by careful selection of which inputs were used for the buttons, they can all be mapped on to the same underlying PIC port so they can all be read with one "PEEKSFR PORTC_SFR, b0" command which would make timing accuracy more consistent.
So a 20X2 is what I would choose.
Hardware-wise; 7 x buttons with 7 x pull-down resistors. An 8-LED+ APA102 RGB LED strip. A 4-digit 7-segment display module, probably a MAX7219.