We sometimes say something like "the word 'love' is a verb". But this is at best a shorthand, at worst a misunderstanding. "Verb" is a part of a sentence, and until a word is used, it doesn't become a verb, noun or any other part of speech.
This is relevant since it is the use of the word that determines which part of speech it is. Now the word "love" when used in a sentence "I love chips" is indicating a state. In this way it differs in meaning from "I eat chips", which indicates an action. This is a semantic difference, not a grammatical one.
Now the present continuous can be formed, and it indicates a continuous or repeated action. "I am eating chips" is a continuous action. But semantically "I am loving chips" is slightly odd. It is odd in the same way that "Green ideas sleep furiously" is odd. The grammar is fine, but the meanings jar.
We interpret "I am loving chips" to mean that the state is temporary. That is an odd idea. More natural would be "I'm loving these chips", because it is understood that after they are eaten they are no longer loved.
However, in formal contexts we want to avoid this kind of semantic oddness. And we usually can, for example the word "enjoy" can be used to indicate a state or an action, and so "I am enjoying chips" doesn't cause problems.
Over time, such differences in meaning can become differences in grammar. Ancient Indo-European had different grammar for "things that are alive" and "Things that are not alive". This was based on meaning. Over time this became fixed as the grammatical genders "masculine/feminine" and "Neuter", that anyone learning French, Spanish or German (for example) will know.
So when someone writes "I have been loving you", the word "love" is coerced from its usual meaning of indicating a state, to indicating a temporary but continuous action. As such it is rather dramatic! The speaker is saying that their love is not permanent. As such it is acceptable if that is the meaning that is intended.