Are you looking for a platform or hosted VMs with a database software running?
I'm wondering how you want to compare SQL Server v PostgreSQL as well. Aren't there different costs for development or admin across these? If you have staff and don't care, PostGreSQL should be less licensing, so lower cost.
Both can work well, or work poorly. You need to consider the cloud platform and how latency affects you. AWS is more mature, and some solid systems run on it every day (Netflix, etc) and it seems to have fewer outages. I find Azure to work great sometimes and be flaky others, but I've run across customers that love it and find it to work very well. I think if you pay for performance, then at larger sizes of VMs/DTUs/whatever, it can work well.
The MS platform has lots of tooling integration. PostgreSQL is mature, but I think the AWS tools can seem raw for an MS stack developer. Again, staff costs are something to consider.
Cloud pricing seems to vary dramatically, and changes quarterly. I think AWS is less expensive, but that can change.
It's a hard choice. I think with either of these, you have lots of flexibility, but you'll burn time/salary to do whatever you want. Ideally, you want to fit within their paradigm closely to limit the development costs and efforts.