Unfortunately there is no available feature to open an application in a new window or a different browser. This idea has been request here: https://support.okta.com/help/ideas/viewIdea.apexp?id=0870Z0000007eeM (https://support.okta.com/help/ideas/viewIdea.apexp?id=0870Z0000007eeM )you can go ahead and vote up this idea so our team can go ahead and start a roadmap for this.
I was trying to Vote for this feature to be considered. I have apps that use 2 different databases from the same URL. So I created 2 chiclets. one for my LIVE database and one for TEST database. I would like the TEST database chiclet to be opened in a new window(private/ incognito window) or perhaps a new browser altogether.