Try following steps, maybe something will give you a hint on your problem:
1. Run putty and after setting up server name/user name/keys etc. save those settings.
2. Run plink.exe or plinkw.exe like plink.exe user_name@server, (it'll throw some info at you) just to see if plink can connect to the server.
3. Check again that git knows that it should use plink, I'd say with all VCSs, that's the problem people have most often, apart from their keys not being set up properly.
Normally by now most of your ssh problems would be revealed. Now just fix them. :)
Update:
I think the problem you have is caused by several gits you've got installed. For the msysgit and cygwin git you'd need to set the GIT_SSH variable. That's done via either 'set GIT_SSH=c:\path\putty\plink.exe' or 'export GIT_SSH=/your/path/putty/putty.exe' respectively. If you're also using tortoise git, you'd need to locate 'properties' (or is it settings?) in it's menu, and set the ssh client there.
Apart from all that, when specifying git clone URL, use your login name and the server name in there, e.g. 'git clone ssh://user@server/your/path/repo.git User and server names should be used by plink, and plink should be able to connect to the server with those arguments, when all of those requirements are met, you'll be all set.
Good luck.