While InDesign and Illustrator don't have access to the Apple feature to show special characters, you have an even better feature in InDesign and Illustrator—the Glyphs panel. It lets you view every character (glyph) in every font on your computer. You can insert a character by double clicking it. In InDesign, choose Type > Glyphs to open it up. Use the font menu at the bottom to switch fonts. Choose from the Show menu in the middle some collections of special characters for different purposes.
Yes, I do use the glyph panel now, but that requires clicks to open it, and sometimes I have to scroll thru many pages of it to find the character I want, depending on what font I am using. It would be so much easier to just use the keyboard feature first and only open the glyph panel when the keyboard feature didn't have the character I want. Fewer clicks = more productivity ... I'm talking a 1,500+ page document here, I need all the productivity I can get ....
Did you notice you can save glyphs (recents) and create Glyph Sets?
The Apple feature will not be in InDesign. It's platform specific.