The oil isn’t eaten up by the lye exactly. One molecule of, say, olive oil, is a triglyceride. It’s three fatty acid chains and a glycerol backbone linking them together. The sodium hydroxide breaks the bonds and the sodium binds with the fatty acid to make the sodium salt of that fatty acid. In the case of olive oil, most of those fatty acids are oleic acid. The glycerol backbone combined with the hydroxyl group becomes glycerin. So depending on the oil and how much of each fatty acid it contains, you’ll have different fatty acid salts at the end - different salts give different results when lathered.
Edit: here is a good video to watch to understand the chemical reaction. This is, however, NOT a great soapmaking tutorial and I wouldn’t recommend you follow his process or safety precautions. He is a chemist, not a soap maker. Some of the things he does and uses could be dangerous.