The character of Mr Game and Watch was created by Makato Kano. In an "Iwata Asks" interview from April 2010, Satoru Iwata describes him as being
... involved with every aspect of design, from the character known as Mr. Game & Watch to the exterior box.
A later section discussed the design of the LCD graphics:
Iwata: After an idea came together, ...
The processors used by Nintendo Game & Watch, are Sharp SM510 or SM511 microcontroller with embedded ROM, that are indeed used also in some calculator.
MAME emulates all the Game & Watch (a lot of them) of which ROM program has been extracted.
Further reference: Emulation General or MAME emulator handheld source code