It's the oblique case. As AJ72T says below, the sentence implies coming TO the house (even if the word को is not explicitly used). Anytime there is a postposition (to, in, on) the noun switches to the oblique case.
My house is red: मेरा घर लाल है
I am in my house: मैं अपने घर में है (oblique)
Does the pronunciation sound to you like घरात , too? Are such liaisons frequent in real life Hindi? I know there is a tortoise pronunciation option in addition to the casual one, but boy do I wish for a third version (at normal speed, but with the speaker clearly enunciating every word boundary, somewhat artificially staying as close to the written letters as possible) or a lesson that teaches how slurs/shortcuts are made at word boundaries.
मेरे घर को could actually be used by native speakers as a colloquialism since 'को' also gives a hint of 'towards' (Even though it is grammatically incorrect). However, this sentence represents a case where the particle 'को' is not used, which happens in some cases of the accusative case. Here "मेरा घर" (my house) is in the accusative case, not the dative, and hence does not require the particle while each instance of the dative case requires it.
Just curious... why do you both refer to accusative and dative cases... Hindi has only two cases, no? Nominative and oblique. The oblique functions like the dative (and accusative?) sometimes. I speak Russian as well, which has six cases including dative and accusative — so I’m familiar with the concepts. Just wondering if I’ve missed something in my Hindi grammar study.
Technically, Hindi has 8 cases just like Sanskrit (Nominative, Accusative, Instrumental, Dative, Ablative, Genitive, Locative and Vocative). However, unlike its parent language where word-endings change depending on the case, Hindi uses postpositions to mark case.
So, in practice, there are only 3 cases you have to worry about when deciding the form of a noun - direct, oblique (when the noun is the object of a postposition) and vocative (when the noun is being addressed).
That said, pronouns do change form depending on the case (Eg: मैं becomes मुझे in the accusative and dative, मेरे in the genitive etc).
It pairs up with दोस्त which is a masculine noun.
However, since दोस्त is also used for female friends these days, a sentence like 'मेरी दोस्त मेरे घर आती है' would not be uncommon in regular usage though I'm not sure what hair-splitting grammarians would have to say about that.