Terminal emulator that copies on select
iTerm2 has this functionality.

It's actually really easy to use and works exactly like you describe - you select your type and then middle click to paste within iTerm.  

iTerm2 has many more features that make it far and away better than PuTTY and the best part is that it's free (as in beer) like PuTTY.



Above screenshot, I have selected the content of my Guest directory.  No clicks or keyboard shortcuts have been pressed.

In the image below, I simply press CommandV and paste it from the clipboard to TextEdit.  This is out of the box functionality - I haven't made any modifications to key combinations within iTerm2.