Good question. It might be because, unlike golf, there is nothing in between putting and doing something else... like driving. It is all putting.... so, repeating the word emphasizes the fact that...
You can putt with any club in your bag. There is no rule that you must putt with a putter; using one just makes it easier to not miss. Pro's have even pitched off greens!
Fergus Muir made a 375-foot putt on the par-3 fifth hole at St. Andrews Eden course for a hole in one.
The Longest Televised putt is by short-game wizard Dave Pelz at Whistling Straits in the run-up...