It's quite true that usually, gifts are physical things given. Note that gift and give are cognate (they have the same origin, so they're effectively the noun and verb versions of "the same" word). There's nothing syntactically incorrect about Dad gave me a trip to Disney World, though. (Except some people might assume he gave me a ticket, rather than actually took me there! :)
But I'd want to know what exactly is the reason1 for including as a gift in the sentence? Presumably it's not to clarify that Dad didn't ask me to pay for the trip. Consider also...
1: My dad let me stay up late to watch the game on TV as a gift
...which I suggest is even more "awkward" than OP's example. Reason being that in nearly all cases where I might wish to attach some kind of reason to such a "parental boon", that reason would effectively amount to a reward (for having done my homework, tidied my room, etc.), so...
2: My dad took me to the amusement park as a reward
...is fine if the context makes it clear what I did to merit being rewarded. Alternatively, if there's no obvious "activity / achievement + reward" combination in play,...
3: My dad took me to the amusement park as a favour
You can find more alternatives by googling synonym reward favour gift. One that might sometimes be more suitable, depending on the exact context, is...
indulgence Cambridge Dictionary
an occasion when you allow someone or yourself to have something enjoyable, especially more than is good for you
...where it might be that the highlighted element above amounts to more than is good for the family finances (i.e. the family can't easily afford the trip, but the child is going to be indulged anyway).
1 Noting the now-revised question text, it now seems pretty obvious OP's context would be best served by...
4: My dad took me to the amusement park as a [birthday] present
...where I suggest the "optional" element should normally be included unless preceding context has clearly established that the reason for the trip is it's a birthday present.