The answer is that you need to check trigger_nestlevel
in your triggers. That is, you need this all your triggers:
IF trigger_nestlevel() > 1
RETURN
The configuration option "nested triggers" is a very old server configuration option. it is available for partially contained databases, since the idea with a contained databases is that you should be able to move it from one server to another without being dependent on server settings. Therefore they made server options that affect code execution database options, but only for contained databases.
Ironically, there is no type of database that is more contained than Azure SQL Database, but there they did not add these options to Azure SQL DB.
I would like you to think a second time if you really want to do this on a general level. I've certainly been in situations where I have had to stop triggers playing ping-pong with each other, but I have had a lot more cases where I needed triggers to nest.