If you’re polyamorous, you should get a flu shot.
Influenza isn’t an STI, but it can be spread through close personal contact, including sex, cuddling, hugging, kissing and holding hands, all of which put you either in contact with bodily fluids or in your partners’ “spray zone” from coughs/sneezes/etc. And if you have three partners, and each of them have 2-5 partners, and so on, one infection can turn into an epidemic.
Some people are unable to take a flu shot, or are particularly susceptible to flu infections/complications. Protect yourself. Protect your partners. Protect their partners.