It is very cute!
Now that we recognize we can use the word "it" to refer to the cat, we can clarify why it does not refer to the general situation by which the cat is being cute. To do so is easy. Analyze the grammatical parts of the sentence:
It is very cute!
It - This word is used as a pronoun. Since we're trying to figure out what the pronoun references, we'll avoid saying much more about it for now. At any rate, "it" is the subject of the sentence.
is - This word is a verb which shows a condition, state, or property belonging to the sentence's subject, "it".
very - This word is just an adverb which modifies the word "cute".
cute - This is the property which is being described as belonging to the sentence's subject, "it".
The question before us, then, is what is meant when we say it? To answer, we can use an old trick that is generally useful for determining the antecedent to a pronoun: replace the pronoun with the nouns we think it might modify. Then decide which expression is more correct according to what we're trying to say.
[It] is very cute!
Or,
[The cat] is very cute!
Or,
[The specific thing that the cat is doing] is cute!
In the last case, the English language offers a convention that can help a listener or reader understand that we're talking about the specific thing the cat is doing and not about the cat itself. To accomplish this, we would use the demonstrative pronoun "that" or "this".
That is cute!
Using a demonstrative pronoun clarifies that we're talking about something specific to the particular situation shown in the image--in other words, the situation. Without the demonstrative pronoun, the implication is that we're referring to a persistent quality of a particular thing, like maybe the cat or the photograph itself.
The adjective "cute", though, modifies the cat, of course. It would sound a bit silly to describe a copy of the photograph as "cute", and it sounds perfectly sensible to call the cat "cute".