Unfortunately there doesn't seem to be much data on the board which I could find.
It does however look like the display is indeed controlled by two 74HC164 chips or similar, so, with those being 8-bit shift-registers, plus 64-segments on the displays in total, I would guess they are wired as a matrix. Send 8-bits of row select data, 8-bits of column select data, the appropriate segment(s) will then light.
As to whether it's 8-bit row then 8-bit column, or vice-versa, or some sort of mix, which are active low and which are active high, would have to be determined by experimentation.
I would guess that sending one of these pairs would light a single segment -
Code:
%00000001 %00000001
%00000001 %11111110
%11111110 %00000001
%11111110 %11111110
Once one segment alone lights, you can then try other combinations to figure out how everything is mapped, which bits mean what.
But you need to be careful because lighting multiple segments could overload the board, burn out the chips. I would ensure you have the board powered from a bench PSU with current limiting turned way down.
The problem is though; this appears to be a 'one segment lit at a time' board, needs to be continually driven to light all the displays one segment at a time. Doing that well is going to be a tough ask for a PICAXE, even at highest operating speeds.
There do seem to be 8 resistors on the board so there is some degree of current limiting per segment, but I wouldn't want to linger on one particular segment being lit without knowing what the resistor values are.
It the resistors are suitable it might be possible to do one digit at a time which is easier for a PICAXE to handle.
You would however be much better off using a MAX7219 display board.