You are correct that the windows version stores it all in the registry. What you have with the .reg file is a great way to copy your settings to another Windows PC. Again you are correct that there is no Registry in Linux systems. Unfortunately nobody seems to have made a tool to read the .reg file that you have and move it to the appropriate config files in Linux.
I had a bit of a hunt and can't find anything. There are some ways to do it from a working Win PuTTY install but nothing to just read the file and get you up and going. The .reg file is just text, you might want to open it up with a text editor and see if you can make sense of it. You might then just be able to copy the relevant fields and keys to the appropriate locations in Linux.