Double-clicks were used with icons on the desktop because you could do more with an icon than just open it. You could move it, copy it, etc. Double-click was a convention for a shortcut to open the reference of the icon. A single-click would have not allowed those other actions.

The idea behind this name for ads was: this company makes ads relevant and compelling, so users drill into them and find whatever you want to advertise.

For what it's worth, because of the affordance you mention, even though users didn't have to, they consistently double-clicked banner ads, and most things they wanted to activate, even after they learned they only had to single click the blue underlined things.

I had to swap real quick to my desktop and see if I still had to double click an icon just to be sure.

I mostly access things with the super key / search now and I guess people with phones would just tap an icon.

Yes it would. And did. Windows chose double click to open but other systems managed with a single click while still allowing you to drag around icons and files.