This one-by-one migration nicely contains problems though if something goes wrong - only one customer affected at a time (or whatever your batch size was) I’ve done deployments this way pretty much everywhere.
On the other hand, I could imagine doing lazy migrations as user or tenants sign in as a nice way to smooth out the increased effort in migrating if you can engineer for it.
I guess per-tenant you would handle similarly but build some tooling to monitor the state of migrations. It might even be easier in some ways, as you might be more able to take table locks when migrating per-tenant.
When updating, schema changes goes first and when successful the new version is allowed to be executed by users. We have small launcher app that the users run which starts the actual application to handle this.
Our DB changes are always backwards compatible in the sense that the old version should operate correctly on an updated DB.
Incompatible changes like renaming a column we handle by rolling out the new column first, then wait till the version still using the old column is retired before removing the column from the schema, or similar strategies.
This way partial updates are safe, as the old app will just continue to use the partially updated DB as normal.
We're migrating to a web app, and hope to have something similar. Plan is we'll update the tenant DB and once done we'll change the app routing so that new user sessions for that tenant goes to new app instances running the updated version.
That said, we've not got 1 million users... yet.
At least with Single Point of Success database design you either move or don’t.
Most systems have a minimum amount of traffic they'll let you pay for, most object stores are not setup to stream small updates in a transactionally consistent way, there's a ton of complexity "in the small" as it were.
Some of other cloud databases don't charge you for database but simply for usage so in that case, usage = customer revenue so cost should scale as you do.