Not safely. For simple sites, probably, but anything remotely complex and the JavaScript would be impossible to deal with.
Lets say the tab has a websocket open, and you clone the tab. What happens to that websocket connection? OpenGL contexts when you're running a game? Both at the same time because it's a browser MMO? CSRF tokens? Any sort of ID that got generated as a message you were typing?
There's no way this can be done without causing things to break in some way. The next best thing is to load it anew, and hopefully 100% cache hit on that.
Good websites duplicate in the blink of an eye because it's 100% cache hit. Crappy websites take seconds to load with a warm cache.