In (iOS && iPhone<9), you can toggle this filter with the triple click trick. In Settings > General > Accessibility > Accessibility Shortcut, select Color Filters, then triple click the home button.
I have a fully updated Nexus - if you search Settings for "gray" you get no results. No "direct access" options under Accessibility. Seems like they may have removed it since 7. They do have some color-blind color correction options, but grayscale wasn't something I could find.
ETA: Ah on the "Go gray" page they describe the steps, including enabling Developer Mode first.