Creating a JWT takes a key or other secret as a parameter, and the resulting token is not superficially human-readable, so it's plausible that a developer might mistake it for encryption based on the high-level "shape" of the API.
Yep. A few years ago I used my credentials in some in-house back-office app that a coworker wrote. Later I was able to see my http calls in the company-wide logging system, with my username and password 'hidden' in a jwt.