I’m trying to develop an external application at app.example.com which is linked on my discourse forum discourse.example.com.
I would like to display the current signed-in username from my discourse instance on the external application. Is there a way to somehow get that username? Like a simple Single-Sign-On with Discourse?
Yep, you can start searching about SSO and/or the API threads related 
It can be possible and I will read the replys here but I think you will need to code something by your own because there is no universal solution to get info from the API and display it outside Discourse.
Something like Discourse embed comments would be super great to communities that have a website next to the forum instance.
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I think you can initiate an api request by external_id to obtain a name on a discourse.