Thing is: if we're looking for free, that may make sense. you still need tinkering though: you need to figure out what the whole dyno thing is about and how to connect from one dyno to another, and so on.
However, if we expand this a tiny bit to cheap and/or free, then simply running the lowest-tier server at Digital Ocean will be a better proposition. And installing a database these days is basically the same thing: run a command to install, all you need is to connect to it. No tinkering. And it will still probably scale significantly better than a free tier at Heroku :)
Once again, I'm biased and old, and can afford to spend money on side projects. I know that sometimes even $15 a month is out of reach (oh, I've been there). But yeah, this sorta prevents me from ever understanding Heroku :)
I... just add it. How do I add it in heroku where everything is "just create an app and push it" and everything is separate "dynos" (whatever that is), each with different pricing, storage, bandwidth etc.