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)
I think there's a mistake in your 2nd sentence:
I am in my house = मैं अपने घर में हूँ (instead of है).
Anyway, it doesn't affect the point you were making. Thanks for your explanations!
The Oblique case or the Dative? There is no postposition, but I imagine this shows direction... homewards? Could you also say मेरे घर को ?
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).
I guessing where you are looking its the abbreviation of "plural".. Could be wrong
You can't use अपने without having previously established a subject. It doesn't inherently mean 'my'. It means something more like 'the possession of the recently referenced person."