It's pretty common in Australia among people who travel for work; get one early-season vaccine and then a booster after they update for the new strain(s).
That might be specific to the Southern hemisphere though; the timing of seasonal vaccine releases is different because our seasons are offset by six months from the majority - I would expect an identical second shot to be much less useful.
That makes sense—it's also true that the efficacy of the flu shot declines over time (maybe 8% - 10% per month?), so there is significant concern about getting it too early. I could certainly see making an argument for getting one as soon as possible and a booster shot in the mid to late season. That's a single shot with a booster, technically, not a two shot series.
I may have found the answer to my lazy question on the CDC website:
https://www.cdc.gov/flu/prevent/misconceptions.htm
"Can vaccinating someone twice provide added immunity?
In adults, studies have not shown a benefit from getting more than one dose of vaccine during the same influenza season, even among elderly persons with weakened immune systems. Except for children getting vaccinated for the first time, only one dose of flu vaccine is recommended each season."
Since they say that "studies have not shown" rather than "we don't have studies that show," I'm more inclined to believe them.
https://www.sciencedirect.com/science/article/pii/S0264410X16310441 doesn't have as many data points in the time series as I would like, but gives you some idea of the drop off. It was enough to convince my parents to get vaccinated later in the year instead of at the first opportunity, but not to get two shots.
I would doubt it—different vaccines provoke different immune responses, and each has a dosing schedule based on empirical evidence about what produces the best response. The fact that two doses are needed for an optimal response from the covid vaccine doesn't tell you much about any other vaccine.
It's possible that two vaccines would produce a slightly better response but they decided the cost/benefit didn't pencil out, and I could imagine that for some immune compromised people, getting two would be appropriate. But I'd stick to the recommended schedule absent strong reasons for doing otherwise.
Caveat: I have a strong layperson's understanding of vaccines, but I haven't looked at data specifically for the flu vaccine.
The dosing schedule often get chosen before the empirical evidence is in. J&J one dose scheduled compared to Moderna's two doses is not based on any empiric justification.
The claim that two doses for a COVID vaccine produces an optimal result is also depends a lot on your goals if your goal is not getting infected more then two are better.
...it works for covid vaccines.