Azure multi tenant saas app, how to register Application per environment?

Any of those help I have a multi-tenant app on Azure not to dissimilar to what you describe. For a largish multi-tenant app I would forget about Web Sites and use Web Roles. I also would avoid having a separate site for each client, they should ideally be running through the same role which can be load balanced over several instances. You can still set up host headers (i.e. site1.mysite.com) for each individual user, all running off the same web role.
I have all my clients running in the same database, this is not to hard to implement and will scale better (you think Google has a database per client?). There is no reason you can't go to more databases later or you can use SQL Azure Federations and have a federation for each client, but going the other direction is much harder. If you have a separate database for each client you will then have to manage each database (schema updates, backups) individually which could be a lot more work (and expense). Cost will influence design here. A 1gb database will cost $10 a month, so if your clients are only paying $20 a month it is not really viable. But if they pay $1000 month then it is not a problem.

like below fixes the issue The Azure AD application roles & roles claims are based on application. We can only create the roles for the application and all the tenants using this application will use the roles defined in the app’s manifest. It is not able to create the tenant specific roles.
Based on the scenario, you may consider authorize the application using the group claims since groups are based on the tenant. You can refer the code samples about Azure AD groups for authorization here.

this one helps. I would say that the third point is rather independent of grails/play/whatever in general. If you need a shared cache there is a multitude of providers for this and there are plugins for most of them in Grails.
The multi-tenancy in grails is pretty mature and much less intrusive than the solution from the blogpost in Sebastiens answer. Whether or not you use single tenancy (multiple databases) or multi tenancy is more or less transparent to your code and most of the headaches are abstracted away. Do be aware that you need to do some smart indexing (like including the tenant id in a multi column index etc) to not get very sad speeds when your data starts to grow.

wish of those help It's a pretty hard question. AAD's multi-tenancy basically requires the org to have an AAD to have proper separation etc.
But in the case of an org not having an AAD, this is one option.

Any of those help Answer context
Before answering your specific questions, let me add some context about the solution I see. With multi-tenancy comes tenant management. And, just like in the document you've linked to, there's probably going to be some sort of catalog holding all tenant-specific information. One of these pieces of information could (most probably will) be a connection string to the tenant-specific database.