I would tend to agree with your professor: a Ph.D. primarily certifies you as being capably of creative contribution to research, and secondarily as an expert in a narrow sub-discipline. Combine that with the continually shifting landscape of the scientific frontier, are there is a great deal of flexibility in what a person with a Ph.D. may end up doing over time.
I have heard one of my close colleagues say that: "One way or another, in ten years time we can't be doing the same thing we are now. Either we will have succeeded and need to move forward, or we will have failed and need to try something else."
In such changes, there is usually a significant degree of continuity that allows one to "pivot" from one area for another. Like in your professor's example, there are a lot of ways in which dog anatomy and fluid dynamics are related, and it's natural that an expert in fluid dynamics might well be drawn to the parts of anatomy most relevant to their existing skill.
A nice real-world example of such a radical transition: Tom Knight made his name pioneering networks and computer architectures, then radically shifted into biology. There is a nice interview with him about his history and how he made the transition, which involved lots of re-education but not bothering with the formality of another Ph.D. He's also moved back and forth between industry and academia quite a bit.
That said, I could imagine some transition so extreme that it might require an entirely new apprenticeship, e.g., from astrophysics to medieval French history. But that sort of change would be a rather extreme an unusual example.