It's just the choice to use the moderately informal "tum" form of address.
Also, tumhaara is inflected to tumhaare because there is a "hidden" /ko/ in there.
main tumhaare ghar (ko).
It is not necessary to say /ko/, but it is understood to be there in the grammatical structure. So yes, that is the oblique case :)
The verb जाना is intransitive which means it cannot take a direct object. So, 'तुम्हारे घर' cannot be the direct object of जाना and there has to be a postposition (को) between 'तुम्हारे घर' and the verb. As Ranzo says, this postposition is usually dropped in practice but still understood to be there.
This is similar to the verb 'to go' in English which is also intransitive. However, we have phrases such as 'go home' where there is no preposition between the verb and the noun. In such cases, we say the preposition has been subsumed by the noun and 'home' is not acting as just a noun in this phrase.