I think that there is a very obvious answer to this question.
The opposite of "better" is "worse."
The opposite of "more" is "less."
So, which of these sentences sounds correct: I like dogs worse than cats
I like dogs less than cats?
The answer is clearly " I like dogs less than cats.
"
The sentence " I like dogs worse than cats " would be flagged by any native speaker as being incorrect.
No one would ever say that.
The answer, in my honest opinion, lies in semantics, not grammar.
Think about it deeply; it simply doesn't make sense to say that you like something better.
The word "better" tells you something about the worth of something.
Whereas the word "Less" tells you something about a quantity (less butter) or about the perceived strength of something (less light).
To increase or decrease the force of your 'liking," only two appropriate words are available to you: you like something more, or less.
You can't like something better or worse, because these are words that comment on the value of liking, not its intensity.
To say "I like cats better than I like dogs" means, semantically, that your liking for dogs is somehow better than your liking for cats .
And it's obvious that that is not what most people intend to say when they use "better" in this context.
What they mean is that their liking for dogs is GREATER than their liking for cats, not that their liking for dogs is in some odd way BETTER than their liking of cats.
Dogs may be better than cats, but our liking of dogs can't be better than our liking of cats.
Of course someone might think that it is better to like dogs than it is to like cats.
But there are no "better" ways of liking something.
english.stackexchange.com/users/16523/Holbrook Robinson
More is, to me, the better of the two.
Perhaps you could avoid all ambiguity and instead of: I like Dogs more.
I like Cats better.
you can try:
I prefer Dogs.
english.stackexchange.com/users/2080/typeoneerror
You can use either, both are short versions of comparisons.
I like cats better than dogs, but dogs more than pigs.
english.stackexchange.com/users/914/Picturepocket
There may be a US/UK distinction.
I like cats better than dogs ...
sounds like to an Americanism to my British ears.
I'd be more likely to say: I like cats more than dogs.
However, I'd agree that both are acceptable.
english.stackexchange.com/users/303/Steve Melnikoff
Historically, like well and like better were the norm.
(You'll find it in Shakespeare.)
British English has dropped this usage, and opted for the more consistent like more , but still retains like well in expressions like well-liked or best loved .
American English has retained like well .
english.stackexchange.com/users/16205/Pitarou
Mike can't like Edie better than me.
It's from a native speaker that seems to be right.
https://www.thefreedictionary.com/prefer
Be Careful!
Don't use any preposition except to in sentences like these.
Don't say, for example 'I prefer art than sports'.
Prefer is rather formal.
In ordinary conversation, you often use expressions such as like...better and would ratherÉ instead.
For example, instead of saying 'I prefer football to tennis', you can say 'I like football better than tennis'.
Instead of saying 'I'd prefer an apple', you can say 'I'd rather have an apple'.
english.stackexchange.com/users/287109/Zhang
When people like something  more  than something else, it's common for me to hear them say they like it  better  than something else. Is this proper English? I've always thought the word  more  fits better, but I'm not sure whether or not use of the word  better  is wrong.
"Like something more" or "like something better"
english.stackexchange.com
<word-choice><grammaticality><adverbs><comparatives><best-most>
english.stackexchange.com/questions/4988
english.stackexchange.com/users/2077/senfo
The answer to this question may be related to semantics rather than grammar and there may be a US/UK distinction. Some people will claim that "more" and "better" are both correct, whereas others will claim that "more" should be the preferred option. It may be that both are acceptable. The answer is obvious, the opposite of "better" is "worse" and the opposite of "more" is "less". The answer relates to semantics and not grammar.
The answer to this question may be related to semantics rather than grammar and there may be a US/UK distinction. Some people will claim that "more" and "better" are both correct, whereas others will claim that "more" should be the preferred option. It may be that both are acceptable. You can use either, both are short versions of comparisons.  There may be a US/UK distinction.  Historically,  like well  and  like better  were the norm.  British English dropped this usage, and opted for the more consistent  like more , but still retains  like well  in expressions like  well-liked  or  best loved, whereas American English has retained  like well .
Both expressions are acceptable and it is a matter of semantics. However, some prefer "more" rather than "better". The answer to this question may be related to semantics rather than grammar and may depend on the regional variant of English. Some people will claim that "more" and "better" are both correct, whereas others will claim that "more" should be the preferred option.
More  seems to be the better of the two. But you can use either, both are short versions of comparisons.   There may be a US/UK distinction.  Historically,  like well  and  like better  were the norm.  British English dropped this usage, and opted for the more consistent  like more , but still retains  like well  in expressions like  well-liked  or  best loved, whereas American English has retained  like well .
Some people believe that "more" is correct and "better" is incorrect in this context. The answer to this question may depend on the regional variation of English. Some people believe that both are acceptable. The answer to this question may rely on semantics rather than grammar.