Hi,
Quite possibly, but perhaps not the way you think. It's a long time since I wrote graphics to that type of display (directly in asssembler), but....
The screen's character display is probably not changing at all, he's just re-loading the 8 user-definable characters. That potentially needs 7 x 8 = 56 bytes to be updated, perhaps 20 times a second, so around 1 kbytes/second. That could even be done with a serial interface (maybe even the best approach with a PICaxe, using hserout) although he appears to be using a parallel interface.
I don't know if the "image data" is being generated "on the fly", but for a PICaxe you'd pull it all out of an external I2C EEPROM which had been preloaded (either by another PICaxe program, or by other means). So perhaps it can be done with an 08M2 ?
Cheers, Alan.