Theoretically they do. Practically I couldn't even get an account last time I tried (a few months ago). It would attempt to charge my credit card a 1 dollar "check the payment is working fee" and fail (even though the charge was going through successfully).
They offer 2 free Oracle databases. The catch is they're Oracle databases.
In other words it's the same as every cloud provider: They hope you'll build a business system that uses enough of their proprietary tech that you can't easily move to another provider.
You can run mysql on any other free database on the 2 free compute instances. It is totally up to you.
The autonomous database is Oracle's offering of their Oracle database - that requires no DBA or tuning and runs on Exadata infrastructure.
They are offering it for free - because they want enterprises using existing on-premise Oracle databases, to move their existing DB applications to the cloud.
Oracle Autonomous databases comes with free Oracle APEX - which is a low code / no code toolchain.