You cannot get Azure SQL to change its version just like you couldn't get a SQL Server 2019 to return a different value. If an app is hard locked to a version of SQL then I'd say the app is broken and needs to be fixed. I would contact the app manufacturer and ask about it. Most likely they have an updated version.
Personally I think you should either get a newer version of the app, migrate to a different app that has better upgrade policies or stick with the existing app and not use Azure SQL. You could still put the DB in the cloud by setting up a VM (lift and shift in the cloud world) that hosts your SQL Server 2016 instance and then have the app talk to it. That is really all Azure SQL is. But you are responsible for keeping the VM and SQL up to date, ensuring there is a database backup and managing the DB instance so you've lost all the benefits of moving to the cloud in my opinion. Still, this would be a step towards migrating to a pure cloud DB.