Doesn't look like that to me. I don't see "firstname.lastname@example.org". IYes, in most browsers the context menu (right mouse click) will let you copy that as an email address.
I don't fully understand the comment about "your browser", especiallyRead that as "the browser you are using".
The application or web site to launch when you click on a "mailto:" link is generally configurable. A web search for "how do I make Gmail my default browser in Firefox" (or whichever email service and browser you're using) will likely turn up instructions.
That said, I don't know if settings like that might be locked down on a shared computer.