What we call the flu is not a single virus that comes once a year. It's different types of virusses each time, and the flu vaccine targets the most likely candidates of that year. So no, you don't generally get REinfected with the flu, because it's a different virus. Some of the virusses that we consider 'the flu' are actually corona virusses, and perhaps covid19 will join that list as it becomes endemic.
There are four other endemic human coronaviruses which typically cause common cold symptoms. Those symptoms are clinically distinct from the flu in most cases.
What medical practitioners call the flu is a small subset of influenza viruses common in humans, each with a high potential for reinfection, including reinfections within a year (as well as constantly evolving variant strains, just like COVID)
Same goes for the T-cell immunity really: it was pretty much a given that the body's normal immune processes still applied to coronavirus, but what was novel was the unscientific community insisting that they couldn't possibly be reinfected and a course of vaccines couldn't possibly give them any benefit to their immune system after they'd been infected (someone should tell all the old people who get flu shots every year!)