Hi Ben,
From what I understand of your context a test manager is the manager of testers. that is not a universal construct. Some use it along with an Engineering Manager (a manager of engineers). It seems to me to be mostly in agile software development shops.
My experience based in northern Europe with many public, regulated, and enterprise companies is that Test Manager is a role similar to a specialized Project Manager. If the delivery has a project manager, then for a certain size a dedicated person needs to lead the testing. Could be what some call a test lead … (?)
What I see in my shop is that people refer to a Line Manager, who then refers to a Director. A line manager can have all kinds of people in their org: Senior Testers, junior developers etc. The org is more and more a stream team in the Team Topologies sense. The hierarchy is for org structure and delivery, not split by capability - say all testers refer to the same manager.