However, "get sick" is a very vague term. In my case, to be specific, about 3 through 9 hours after the shot with a new strain in it, the shot site is tender and I run a small, but noticeable, fever kind of thing, barely worthy of the term but definitely a bit of an elevated temperature. Then it subsides. It is basically a minor inconvenience. By contrast, while I don't get the flu every year, it can hang with me for two weeks and knock me out cold for a good three or four days in the middle. And the shot generally covers more than one flu.

So yeah, both things may be "get sick" in some sense, but just because they both fit into the phrase "get sick" does not mean they are particularly comparable.

(Incidentally, over the past 10 years I've managed to do a lot of improvement to my immune system for various reasons. When I got the shot with a less-strong immune system, I generally didn't notice any symptoms from a flu shot at all. Everyone's different, but I'd suggest you generally should notice some symptoms from an effectual shot.)