How can I open a new Finder window on every click on the Finder Icon?

This behaviour happens beacuse the dock just launches an app. Finder is classified as an app, so it will open/show the app window, you will need to open a new window within the app.

You can use the keyboard short cut CMD + N, and if you are not in the app press CMD + Tab to open it first.

You could possibly try using Quicksilver, which makes custom keystrokes, so you could assign O to opening a new finder window

You can do this using Hyperdock.

I do this for most of my software (Finder, Terminal, Chrome, etc...). Clicking on the application icon sends the application a Command + N.

It's non-free though.