Is a Mechanical Engineering BSc who ends up working as mechanic on a car shop an engineer?

Yes, although those fellas mostly don't pretend their jobs involve designing and validating mechanisms with any scientific and industry regulation / standard rigour

Since you didn't specify the job_title involves any actual engineering, I'm gonna assume it doesn't... just like most job_titles

I too have a piece of paper saying I'm a Computer Engineer BSc (didn't apply for any of the electronics/CE licenses back home because wages were shit and the license wouldn't have been worth anything overseas)

Half of SEs consider the idea of written requirements needless bullshit that just drag you down. Validating your designs with any sort of rigour just isn't there for the vast, vast majority of jobs. As such, calling it "engineering" is a joke

If the activities of your regular Software Engineer can be said to be engineering, so should be changing a tyre. There isn't nearly enough rigour in the process to warrant calling it "engineering"