So it turns out that "having the car smogged" is a California regionalism?
California's emission tests, which you have done before renewing registration, are called "smog tests" because they started as a means of reducing smog. Pollution controls (including, but not limited to, smog reduction measures) are why on most days now, I can see the Sierra Nevada and Coast Range mountains from my spot in the center of the Central Valley. In my childhood, seeing the mountains on the horizon was an unusual and exciting phenomenon. Apparently, we were breathing toxic soup back then.
"Taking the car for its smog test" quickly evolved into "taking the car to be smogged." I've used the term for my entire car-owning life. Authorized smog test centers enthusiastically refer to what they do as "smogging." People in the California subs on reddit use the term.
So I was startled when I mentioned to a colleague in another state that I'd been having the car smogged, and they had no idea what it meant. I used to live in their state; I have taken a car to the big drive-up center to be tested (that car failed because it needed a new gas cap). I took my car for the $15 cash-only tests in Connecticut, too. This is apparently "emissions testing."
How very formal.