You should always run mysql_upgrade to be safe, and I'd assume Uber's automation does this. But you probably won't encounter problems if you don't.
mysql_upgrade typically just makes some quick changes to the system schema ("mysql" database), not to other data. And it typically only needs to change things across major version upgrades (5.5->5.6) or pre-GA releases; it's rare to make a change in a point release of a major version much after the GA release.
That all said, I don't ever advocate using Docker for MySQL in production at this time; see my longer comment in another subthread below for reasoning.
My experience with this mirrors the parents. The more data you have, and the more obscure engine features you use, the more likely it is to happen. It only takes one major failure to make you want to test the daylights out of any upgrades or downgrades.
The more common classes of problem are performance degradation, config options being added or renamed, new features being buggy in rare edge cases, etc. Not things particular to InnoDB's storage format or things that would prevent a volume from being usable with a different version.