For example, if I were to shiny hunt the HGSS starters. On the menu, I get a shiny Tododile, but I select Cyndaquil. Would Silver’s/Rival’s Tododile stay shiny?
NPCs can only have a shiny if it is preset to be shiny. And since it is not preset for your rival, so no, it will not be shiny, just like sumwun said in the comment.
Source Hope I helped! Have a great day!