I don't know that they are synonyms. Most automobile questions probably are a subset of automotive engineering, but I don't think that's necessarily explicit, because I'm not sure I consider the wiring system for your radio and speakers to be automotive engineering, I'd classify it more as electrical, but it's definitely related to automobiles, because they do have a semi-standard setup.
I was going to cite Wikipedia here, but the more I dig, the more I think your second footnote is correct, their arrangement and definitions of automobiles, automobile engineering, automotive engineering, and car is inconsistent and screwy. To me, an automobile is a car, truck, or SUV. It's a small-ish vehicle that people use for personal primary transportation. Something you buy off a lot or in a showroom. When you say "automobile engineering," I think of the design and manufacture of these vehicles. When you say "automotive engineering," that means something much broader to me.
Some of the people I work with belong to SAE, and we have lots of SAE references around (which I'll admit does not make us an automotive company), but we generally don't deal much with passenger cars. Our primary focus is on agricultural machinery, and we have sister companies that deal with construction and mining equipment as well. We also do business with low-volume specialty vehicles; things that are certainly related to cars, but wouldn't fit your typical definition. These are all parts of automotive engineering, the principles are very similar, it's the application and the requirements that set them apart. But when someone says "automobile," I don't picture a Deere tractor. I'm more likely to picture a Camry.
So my feeling is that the tags might commonly overlap, but that they definitely don't cover the exact same ground. I think automobiles is a subset of automotive-engineering, but that it has some distinct uses and can still be useful separately. Especially because I could easily see a large number of questions over time about peoples' personal automobiles, e.g. features and technology present on them, since it's something so many people interact with so frequently.
[structures]. This is an issue with "discipline categorization" tags but I don't think ~300 questions is enough data for us to discern what sort of use pattern will ultimately be best, yet. $\endgroup$