Former influenza epidemiologist here and the answer is probably not. Our current influenza vaccines in the US (and in almost all countries) are trivalent, meaning they have 3 strains: an influenza A H1N1 (aka swine flu), an influenza A H3N2, and an influenza B (Victoria). Fun fact, they used to include 4 strains, but COVID actually wiped out the second B strain and it hasn’t been detected since 2020.
Unfortunately, the current bird flu is H5, a separate influenza A strain. Typically there isn’t a lot of shared clinical protection overlap between genomes this different. This is why influenza vaccines typically use 3 (and formerly 4) different strains of the virus, to confer the most protection.
However, unless you work in dairy or poultry, I would not worry. I am not particularly worried. I know it’s making headlines, but only 66 human cases are in the US so far, and all but 2 had direct contact with those animals. Currently there is no documented person-to-person transmission and certainly nothing like what we saw early on with COVID.
If you want my personal take, I still mask at large events / crowded places / airplanes to avoid flu and COVID altogether. I still encourage everyone who is ill to mask and stay home, even though it’s out of fashion. I think we will see egg, dairy, poultry, and beef prices rise with H5, as farmers are forced to cull animals. Also, we may see cases from raw milk and backyard animal husbandry - please don’t drink raw milk, pet cows, kiss chickens, touch dead birds without gloves, etc. right now and you probably won’t get bird flu.
Hope that helps, happy to answer any other influenza or respiratory infectious disease Qs!
Edited to add: The fact we have human cases at all is because farm workers - usually immigrants with limited English - are routinely exploited, denied PPE, or put at risk in unacceptable ways. Farms are basically self regulated in the US, though they are officially regulated by the USDA. It is extraordinarily difficult to get farms to cooperate with pandemic preparedness. You have to pay them federal money to do anything. There have been documented cases of farm workers being told to cull infected birds with zero PPE, and that’s unacceptable. These vulnerable people are the ones paying the price for our H5 knowledge right now, and it’s not right. It’s disheartening to have worked on pandemic preparedness for decades, fumble COVID altogether, and still fail to prepare for the next one. As climate change continues unmitigated we expect to see more human-wildlife interaction and more zoonotic diseases. We can’t just do good science; we also must address workers’ rights, capitalism, and climate change.