I personally have Caprover running on a $5/month DigitalOcean droplet where I can create DBs on the fly for small projects. I normally switch to a better hosted platform for bigger projects. Heroku is also good but the free plan is 10,000 rows and the next plan is $9/m.
I don't think there's an official recommendation. Any database host will work just fine. However if you want to optimize based on location, here are the regions where Vercel hosts: https://vercel.com/docs/edge-network/regions#routing. So hosting a database in AWS in any of those regions probably makes the most sense.