Every vaccine you have ever received is, linguistically, a cow product.
The story starts with smallpox. Before Edward Jenner, people were protected by variolation, deliberately infecting a healthy person with a mild dose of smallpox. Practitioners in China blew powdered scabs into the nose, with documented use in the 15th and 16th centuries. Lady Mary Wortley Montagu brought the practice from Turkey to Britain in 1721. It worked, but it carried real risk.
Jenner found a safer route. In 1796, he took material from a milkmaid's cowpox sore and inoculated an eight-year-old boy. When the boy was later exposed to smallpox, he did not catch it. In 1798 Jenner reported that his method was safe and could be passed from arm to arm. He called cowpox variolae vaccinae, Latin for "smallpox of the cow," from vacca, "cow."
English picked up the adjective vaccine in 1798, meaning "from cows." Jenner himself used vaccination for the procedure by 1800, and vaccinate followed in 1803 as a back-formation. The noun vaccine, meaning the matter used in vaccination, is recorded from 1846, via French vaccin.
For decades the word meant only the cowpox preparation. Then the science moved on. Around 1880 Louis Pasteur developed protection against chicken cholera and anthrax, neither of which involved cows. In August 1881, at the International Medical Congress in London, he proposed extending the words vaccine and vaccination to these new inoculations as a tribute to Jenner. Etymonline credits Pasteur with the broader use too, though it does not give a date.
That is why a flu shot or an mRNA injection carries a word that points at a cow. The name outlived its origin by about two centuries and shows no sign of changing.
Even the root is a mystery. Etymonline lists vacca as being of uncertain origin, so the history of the word is clearer than the history of the animal's name.