This is indeed an object with a copula but not a direct object (leidend voorwerp). If it were a direct object you should be able to turn it passive, which is not possible. This type of object is called an oorzakelijke voorwerp in Dutch (causal object), even though in many/most cases there is not causal relation.
Note that it doesn't have to be "het": you could say "Ik ben jou beu" (I am sick of you), or "Ik ben het werk beu" (I'm sick of the work).
There are several similar expressions:
- Ik ben het kwijt. (I lost it.)
- Ik ben je vier euro schuldig. (I owe you four euro.)
- Ik ben iets van plan. (I am scheming/I am planning to do something. (not a perfect translation))
- Ik ben het met je eens. (I agree with you)
- Dat huis is het waard. (That house is worth it.)
This last example is the same structure as in English, so there should be a corresponding English term, but I don't seem to find it on Google.
Concerning how something like this might evolve, is seems like (see here) the phrase also existed as "Ik ben beu van ..." (which, incidentally, is the same structure as the English translation "I am sick/tired of...").
This website suggests that many such oorzakelijke voorwerpen used to be genitiefobjecten (genitival objects) in Middle Dutch but evolved into voorzetselvoorwerpen (objects of preposition) with the preposition "van" (of). The example given is "sijt des seker" in Middle Dutch becoming "wees daar zeker van" (be sure of it) in modern Dutch. But some did not, and the phrase "ik ben het beu" seems to also have gone in that direction, but then the evolution stopped and the old form continued, be it without a genitive form.