Typically, “have got” is considered the informal structure of “got” (see here for an example). This presumes that the meaning is the same, but the construction has a formal/informal tone (in other words, the deep structure of the sentence is sustained).
This certainly works when the object is a noun. Let’s look at four sentences where this works:
1A “I’ve got a present”
1B “I have a present”
2A “I’ve got a boyfriend”
2B “I have a boyfriend”
But this dynamic change when the object of the sentence is a pronoun. In fact, the two (“have got” and “have”) are no longer semantically similar when we apply it to pronouns. Consider the following three sentences:
3A “I have you.”
3B “I got you.”
3C “I’ve got you.”
In the first of these three sentences, I have you is an indication of possession, and can be somewhat creepy in the wrong context (after all, will you have me for lunch?). However, it can also be used in a relational context. For example, see the sentences below, from COCA:
4 “You have me at a disadvantage” (Fiction)
5 “Will you have me back on the show and apologize in person?” (TV News)
6 “Once you have him chitchatting, he might inadvertently let something slip.” (Magazine)
7 “I can do anything if I have you with me.” (News)
Here, there is still the essence of possession, as typically the subject is in a position of authority. However, in the last example, [7], we can see the use of “have” in the relational context, which is where [3C] comes in.
The sentence “I got you” shows the complexity of the verb ‘to get’, since it holds multiple meanings. With nouns, often the “to get” is also an indication of possession (“I got coffee [for you]”). However, with pronouns, the verb “to get” is a signal of understanding (“she gets me” or “he gets her”).
The syntax of the last sentence, “I’ve got you” is a mix of both—possession and understanding. This combination is so much more meaningful than the individual terms, making the syntactic construction “have got” more complex than the simple “formal/informal” narrative we traditionally learn.
3C “I’ve got you.”
9 “You’ve got me.”
In the four “words” above (technically three words, one of which is a conjunction), we are expressing both possession and understanding—in other words, trust. When I say “he’s got her”, I am indicating that the object (“she”) can trust the subject (“he”) because the subject is capable of taking on (possessing) some burden, and because the subject understands the weight of that burden.
The expression of trust and wanting to be trusted is especially clear in first- and second-person pronouns, such as in [3c] and [9]. It’s the kind of language—perhaps my literal love language—that exists in relationships, when we rely on one another immensely as day-to-day safety nets. But it’s also something I find myself saying to friends and close compadres. When I use [3C], I am expressing appreciation for you by indicating how much I trust you. When I use [9], I am expressing a desire to be trusted.
So much can be said in so few words.