Ha, well, I understand your perplexity as I have now read a thread at another forum over frustration at not being able to delete the Search icon. So how did I do it? Ha, I'm not sure. What I remember is that I right clicked on the icon and chose unpin, something like that. If I remember right I did this with two of the icons in the taskbar within minutes of Windows 10 being installed. If I try to do that now I no longer get that little box. Instead it is the taskbar window that opens up. I wonder if I got an update very quickly after the installation which changed things around a bit. Weird, huh?
I tried your suggestion and I do get the search box if I click on Show search box, but that is not the same as the icon that used to be there (a magnifying glass, yeah?) So, the Search icon was for searching the web? Seems to be what Cortana is for, yeah? So no loss, right? Since many people wish they could get rid of it too, ha. I thought maybe the Search icon was for searching the system, which I still don't know how to do if I want to. That is why I wanted it back, to search the system. There is no search app in the start menu for doing that.
I also seem to remember the Control Panel showing in the Start menu, but then later it wasn't there. It's been rather confusing so I'm not sure now about that. At any rate I figured out it is the Settings icon they want us to use instead of Control Panel and in fact I used it to delete an app I no longer wanted.
So now I am wondering if I am remembering this right....... Next to the Start icon were 4 other icons, yeah? Cortana, Task View, Search, and....... ? Is there another? If so, I can't remember what it was. Can you tell me?