I got three notifications about hats today:
Does this mean I earned the hat once, for well-received answers on those three sites, or three times, for well-received answers on 9 sites?
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You got it for having at least one good answer on those sites. So if you have 1 good answer on each = Polymath. So that's 3 answers. The hat is awarded simultaneously for the three sites.
I have this on
4 5 sites, for having at least one answer scoring at least 1 on each site. Like on ELU, I only had one answer, yet I received the hat.
This means you earned the hat once because you had a well received answer on The Workplace, Personal Productivity, and Travel. The hat is awarded on all 3 sites in which you had a well received answer.
For example, if I had a well received answer on SuperUser, Meta Stack Exchange, and Travel, I would get 3 hat notifications because I got the hat once, but it was awarded to me on the 3 sites where I had the well received answer.