How Do Sites Suppress Pasting Text?

Probably using the onpaste event, and either return false from it or use e.preventDefault() on the Event object.

Note that onpaste is non standard, don't rely on it for production sites, because it will not be there forever.

$(document).on("paste",function(e){
  console.log("paste")
  e.preventDefault()
  return false;
})

Even if it is somewhat possible to intercept the paste event in many browsers (but not all as shown at the link on the previous answer), that is quite unreliable and posible not complete (depending on the browser / OS it may be possible to do the paste operation in different ways that may not be trappable by javascript code).

Here is a collection of notes regarding paste (and copy) in the context of rich text editors that may be applied also elsewhere.