Actually…I think this is like “If I would have known”…that technically it doesn’t work, but it’s very common, to the point that I write it off as a colloquialism.
Here’s why it’s confusing: both “more” and “better” can function as either adjective or adverb. The question is which function is being filled in the case of “I like it ___.” Once we know what part of speech we are using, we can choose the appropriate word to fit our meaning.
What is being compared here? How well I like, or what is liked to a greater degree? If the comparison involves how well (as in how adeptly or correctly or skillfully) I like something, then we are comparing my liking it vs. somebody else’s liking it, or perhaps to how well (adeptly, correctly, skillfully) I like another item.
Otherwise, if we are comparing to what degree something is liked, i.e., I like this sweater to a greater degree than I like that sweater, we have a different situation. I like one sweater not more adeptly, not more correctly or skillfully, but to a greater degree than another sweater.
Let’s apply a different verb to the same concept. If I throw very well, then I throw farther or faster, etc. If I throw a basketball as soon as I get up in the morning and I throw it all day, but I only throw a baseball once, I throw a basketball to a greater degree or amount than I do a baseball, even if I cannot throw the basketball with any greater skill or strength than I can a baseball.
If I throw farther or faster than you, we can say I throw better than you.
If I throw one object to a greater degree, amount, or constancy than I do another object, we can say I throw one object more than I do the other.
So, when I say “I like this sweater ___ than that one,” am I comparing how adeptly I like it, or to what degree? Are you more skillful at liking one sweater than another, or does it please you to a greater degree?
“I like this sweater more” makes sense; “I like this sweater better” does not…but as I stated in the beginning, it’s a big colloquialism, and I don’t expect people to get rid of it soon. (sigh)