Having talked to Tableau Support, apparently it is not (as of 2018-06-08) not possible to connect to PostgreSQL Aurora, only MySQL Aurora. An excerpt from their answer:
"The Development team have responded to my email and confirmed my findings. Currently the Amazon Aurora connector in Tableau Desktop only works with an Amazon Aurora MySQL compatible edition instance. Therefore I will submit a feature request on your behalf to add functionality to the Amazon Aurora connector; so that it also works with Amazon Aurora PostgreSQL compatible edition instances. The Product Management team will review this request for possible inclusion in a future release."
Indeed I missed the fact that you are using PostGreSQL.
Keep an eye out on future releases, they might come in with an upgraded Aurora connector.
I was able to use the regular postgres driver to connect to Aurora Postgres. Somehow it didn't work the first time I tried.