"Engineer" is a protected job title in the US, but most are able to operate under an industrial exemption and so legally use the title.
The reason why some are picky about the title is that it is a licensed profession, and has some expectations associated with it. If someone tells me they're an engineer, I assume they know how to use a Laplace transform, because that's common across disciplines at the undergraduate level. Same as applying perturbation theory at the graduate level. Same as having a common corpus of basic science knowledge. Same as with professional code of ethics. Etc.
To use your own logic above, calling yourself a software ninja doesn't make you a ninja, and calling yourself a software engineer doesn't make you an engineer. It might be legal, but it's crass.