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.
Best Answer
Fond can be considered as more than like and less than love. (Used as be fond of)
Though, it might depend on the context and it can be synonymous to like also. It is usually not used in a romantic relationship and you can both love and be fond of someone. For example, you can both love your niece and be fond of him because you like to spend time with him.
Other than that, infatuation can be considered as an early stage of a true love (used as infatuated with) but it is not necessarily between like and love. It is called puppy love also.
For example, an urbandictionary entry says:
Also, attraction can be stronger than like and weaker than love. (used as attracted to). Though it is usually a physical attraction and you can be attracted to someone you don't like. (an emotional irony?)
In the end, it might be better to use several words to explain the emotion than using a single word that might not be enough. There are different situations and contexts.