There is no “save draft” button, your message will be saved every few seconds automatically to make sure you don’t lose anything if you navigate to a different page or hit the back button by accident. You will find it in /my/activity/drafts afterwards.
I can’t speak for the Discourse team but I think it is designed that way on purpose to encourage people to finish posting a reply instead of ending up with a big list of half-finished replies over time.
The way to start a new draft is just to click reply on a different message than the one you are working on, you should then see this message and be able to click the “no, save draft” button:
But maybe it would be safer if saving the draft was the default action. I usually just work on one post at a time so it hasn’t bothered me 