Does the "cattle not pets" distinction apply as equally to machine instances as to containers?
While technologically, containers and virtual machines are very different, there is no apparent difference from the perspective of your software. It seems like the argument in your question is that data is special and will always be a unique snowflake, so your question basically boils down to what to do about it in terms of DevOps, CI and Automation.

This is the perpetual challenge of the DevOps model. Ultimately, what you are asking is, either A) your data will be in a database or B) a datastore , so how do you manage that?

The answer is that yes, your database servers can and should also be treated as cattle - and I recently detailed several techniques and technologies for doing so in house. If you don't treat your data storage (databases, storage filers) as cattle you will find you have a single points of failure within your data(base) infrastructure and a lack of scalability and redundancy. These techniques I discuss in my linked answer basically allow you to distribute and cluster your relational database in a manner similar to Cassandra.

Managing your data(bases) and making data(bases) redundant is probably the most difficult challenge facing DevOps. The easiest way to solve the problem: outsource the headache to a cloud provider.

So in short, yes the "cattle not pets" distinction apply as equally to machine instances as to containers.