In American English, have got behaves somewhat like a verb. Does this mean it is one? That depends on who you ask. But there is some justification for your textbook's position. However, as the OP says, calling have got a verb might be really confusing in ELL because there are lots of ways it doesn't behave like a verb. But calling it the present perfect might also be confusing for students of American English, because it doesn't behave like the present perfect in AmE, either.
Americans say (or at least some of us do)
I've got the tickets, don't I?
If we thought of have got as the present perfect tense of get, we would say
I've got the tickets, haven't I?
Furthermore, the actual past participle of the verb get (at least, when it means acquire) is gotten, and not got.
However, neither of these reasons for considering have got to be a verb holds in British English. So in British English, it may be better to think of it as the present perfect form of to get.