Is there a way to open an unity application (or any .exe file) from a unity browser game?

I am not sure how Battlefield does it but there are two possibilities.

Either they make you click on a url that has a custom protocol like steam://rungameid/220200 which would lunch Kerbal Space Program through Steam. This method requires to have a program that is registered as opening those URLs in the registry. It has some limitations, the user has to click on the link and there is no communication between both applications beyond the content of the URL. This is probably how Battlefield does it.

The other possibility is to have a NPAPI browser plugin. The problem is that those have become deprecated has they are too much of a security risk. Such plugins run as part of the browser with the same user permissions as the browser does. They can open processes and initiate two way communications between them. It is also then possible to use a window handles' SetParent to have an external application display inside of the browser window, faking being native web.

I have implemented the later method for a previous company in the early 2010s. Although the results were great, there was also a substantial number of users who stopped using it since the browser displays so many warning before installing such a plugin. Now that all major browsers have stopped supporting this technique, there is no point to this anymore.