Good day Jakub,
how can I assure that after region is down my database is still functional?
This is good question and the answer is simple: "Azure Database for MySQ" is a service managed by Microsoft and they guaranty your service as explained in this document:
The Azure Database for MySQL service provides a guaranteed high level of availability. The financially backed service level agreement (SLA) is 99.99% upon general availability. There is virtually no application down time when using this service.
Azure Database for MySQL automatically creates server backups and stores them in user configured locally redundant or geo-redundant storage. When you create the server, you have the option to choose between locally redundant or geo-redundant storage. If you want to make sure you do not lose anything if the entire region fail then you can select the geo-redundant.