Hello @findingbugs, before knowing the dependency you should know what actually is AZ and Edge location
Here you can find the definition of Edge Location and understand the working of Availability Zone.
Coming to your question,
Yes the DB Instance is dependent on Availability Zone. Any DB Instance you create runs only in the Availability Zone.
In order to work in a different region, you can change the region from your console and then again run/ launch an Instance on it.
Hope this clarifies your understanding on Instance running in a particular AZ.