Say for example you find 78 Kabuto without shiny in Pokémon GO, does the odds chance to 78/4096? Or what about if you get 689 encounters without a shiny in UM? Does the chance go up to 689/4096? Is this the case, or is shiny chance just random with the odds?

The shiny chance is just random with odds. If it increased as you are asking, then you would be guaranteed a shiny by the 4096th encounter, (4096 encounters would make the chance go up to 4096/4096, which is 100%) and this is definitely not the case.

The chances are always at 4096, its mainly luck , the more envounters you get USUALLY you will gte the shiny , bit the actual odds are always the same.

Source:knowledge and experience