There is η οικία which is (very) formal for house and ο οίκος which is the word used for an important family, a dynasty or a company or a social welfare institution.
The everyday, general word for house is σπίτι, yes, unless you talk formally. For example, in the news, you will hear Η συνάντηση θα λάβει χώρα στην οικία του προέδρου=The meeting will be held at the president's house, and not στο σπίτι του προέδρου.
Just because it came up in my mind: Σπίτι and hospital share the same ancestor "hospitium" from Latin.
So σπίτι is a loanword from Latin? When did it enter the Greek language?