What's being said is that virtually all animal products, even the leanest cut of meats and birds have some fat. There are some shellfish that are close to having no fat, but not sure as to whether you eat those. I know that bottom feeders have a negative connotation for some people.
I'm wondering if in the bible, they were referring to visible fat. I'm so clueless as to bible stuff because I withdrew at an early age.
"Right is right, even if no one is doing it; wrong is wrong, even if everyone is doing it." - St. Augustine