THis is the key of the house.

AwenceThis is the key of the house.

It's correct. You can also say:

• This is the key for the house.
• This is the key to the house.
• This is the house key.
• This is the house's key.

i m so puzzled ro see this sentence appear in the "error identification" section in the exam.

Awence"error identification"

The default sentence is "This is the key to the house", but don't feel anything is wrong with your sentence.

Nobody says "the key of the house". It's "the key to the house" or "the house key".
CJ

CalifJimNobody says "the key of the house".

KJV: Isaiah 22:22 [A]

This is the house's key.

Although it not wrong, I find it rare that people would actually say that.

GoodmanAlthough it not wrong, I find it rare that people would actually say that.

Why? Because a house is inanimate? Or is it something else?

It's just the way it sounds to me, a at least in this nick of the woods.
It has nothing to do with being inaminate.

HuevosIsaiah 22:22

Well, if you wish to seem as ancient as the man who wrote that, please feel free to use that expression!

CalifJimthe man who wrote that

That text was prepared by 47 learned scholars from the University of Oxford, the University of Cambridge, and the Palace of Westminster (or should I say Oxford University and Cambridge University?). Anyway my point was that there is nothing grammatically wrong with the construction.