I'd probably pick PostgreSQL over either of the other two today for any case where my constraint was that I had to have the application run on Linux. Otherwise, I'm going Azure SQL Database or SQL Server if on-prem (as in, non-cloud) is needed. This will change soon with the OP about SQL Server on Linux.

SQL Server Management Studio is an incredible tool and beats the pants off of equivalent free tools like MySQL Workbench and pgAdmin. (SSMS is now freely available without an MSDN subscription or SQL Server license.) Yes, it requires Windows, but compared to the other tools on Windows it is second to none.

Azure SQL Database has been nothing but a good experience to work with. You automatically get 3 highly-available replicas of your database for no extra cost, built-in transparent data encryption, query auditing support, threat detection alerts, near 100% compatibility with SQL Server, and no need to keep the underlying OS or software up to date and secure. All of this for $30/mo (Standard S1 we've found to be adequate for most small businesses with line-of-business type apps.) Depending on the needs of the application, Azure SQL Database can either be a good replacement for an existing SQL Server instance, or a stepping stone towards a larger SQL Server installation. We're seeing many small businesses power down their on-prem/colo SQL Server installs and moving to Azure SQL Database to save on TCO.

SQL Server Express Edition is a free edition that supports up to 10GB databases, so it is perfect for your local development environment (assuming the size constraint allows that). In 2016 SP1, they even added all of the Standard and Enterprise edition features to Express edition like memory-optimized tables and columnstore indexes. With integrated Windows authentication, it is incredibly easy to have your team use a single connection string and for a new dev just install SQL Express and go.

SQL Server Database Projects in Visual Studio are by far the biggest reason for me to prefer Azure SQL Database or SQL Server. This is the most elegant way of storing your database schema in source control that I've ever seen. You define the schema as just CREATE scripts, and when you build you generate a DACPAC with the entire schema defined as if you're creating it from scratch. But upon deployment it determines what objects need to be created, altered, or (if you want) removed to make the target database schema match the normative schema in the DACPAC. So it builds its own migration script based on the schema of the target database, and it won't allow any operations that cause data loss by default. This is super handy in a team environment because you don't need to worry about writing migrations by hand, you just define the schema "as it should be" and no matter what version of the database your team members had on their machine it will get caught up. Also dealing with merge conflicts is easier, because you're just doing a line-by-line merge of the e.g. CREATE TABLE statement rather than having to worry about which order your migrations run in. If anyone knows of something equivalent for MySQL or PostgreSQL I'd love to know!