"Ask me my name."
Translation:मुझसे मेरा नाम पूछो ।
It makes sense to me if I think of it as "ask of me," like I'm going to give you my name in response, it's something that will come from me. Idk.
This is an Imperative Sentence. The sentence "Ask me my name" can be logically written as "(You) Ask me my name"
So, the Hindi Sentence is:
(तुम) मुझसे मेरा नाम पूछो।
अपना cannot be used here.
If we said (तुम) मुझसे अपना नाम पूछो। we would be asking "(You) ask me your name."
The meaning changes.
That would change the meaning totally. If you put apna in place of mera, the sentence would then mean - Ask me your name.
You can find some examples of usage here: