Yes, the common root is french, as in "Normans" :)
Meat from cattle is beef. Steer in French is beof ( https://en.wiktionary.org/wiki/beef : From Middle English beef, bef, beof, borrowed from Anglo-Norman beof, Old French buef, boef (“ox”) )
Meat from a chicken is poultry. Chicken is poulet in French. ( https://en.wiktionary.org/wiki/poultry : From Middle English pultrie, from Old French pouleterie, from poulet, diminutive of poule (“hen”), from Latin pullus (“chick”). )
Meat from a swine is pork. The word for swine in French is porc. ( https://en.wiktionary.org/wiki/pork From Middle English pork, porc, via Anglo-Norman, from Old French porc (“swine, hog, pig; pork”), from Latin porcus (“domestic hog, pig”).)
This is because when the normans (who were the rulers at the time) wanted poulet on the table, they didn't want a live chicken - they wanted a cooked chicken and so the word the meat and the animal diverged in English.
There are also some interesting Spanish / Arabic word pairs from https://en.wikipedia.org/wiki/Muslim_conquest_of_Spain where the word in Spanish differs from the romance side of the family tree.