I'm using msysgit running on Windows XP.
Tried Ctrl+V, Right click, Middle click, google... no luck.
Press Insert.
Also, to copy from the window, try clicking the console's window icon (topleft) and choosing Edit -> Mark, then drag a box on the text, then press Enter. (You can also paste via the window icon menu, but the key is faster.)
UPDATE
Starting from Windows 10 the CTRL + C, CTRL + V and a lot of other feature are implemented in conhost.exe so they should work with every console utility on Windows. (You have to enable Properties -> Option tab -> Quick Edit Mode)