Shiny statues are called "color statues" by the games themselves. The name "shiny statue" is a misnomer; colour statues do not affect shiny rate.
Colour statues have the same effect as regular statues, where the encounter rate for Pokemon with the matching type in the Grand Underground is increased. Colour statues have a stronger effect than regular statues.
To increase Shiny odds underground, you need to fill the Diglett meter, which will temporarily give you an extra roll (effectively doubling your Shiny chances).