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... He scowled at the dark ceiling. Did they think he couldn't look after himself? He'd escaped Lord Voldemort three times; he wasn't completely useless....

Unbidden, the image of the beast in the shadows of Magnolia Crescent crossed his mind. What to do when you know the worst is coming....

“I'm not going to be murdered,' Harry said out loud.

'That's the spirit, dear,' said his mirror sleepily.”

I don't see a real mirror has been mentioned in the whole context. No one else was there and Harry was lying in his room alone. It might not be a literal mirror here. What does "his mirror" mean?

-- From Harry Potter - The prisoner of Azkaban Chapter 4, the Leaky Cauldron.

It's not a figurative mirror - there is a real talking mirror in Harry's room at the Leaky Cauldron which is mentioned earlier in the chapter.

He caught sight of himself in the mirror over the basin.
...
'You're
fighting a losing battle there, dear,' said his mirror in a wheezy
voice."

The world of Harry Potter's books is full of wizardly items that include talking portraits and magical mirrors. Probably it was a mirror like the one depicted in Snowwhite's tale, a talking magical mirror showing Harry's image but with his own voice and thoughts.

You seem to have some doubts about the use of his. Let's check the dictionaries

his

belonging to or connected with a man, boy, or male animal that has just been mentioned or is known about

belong

synonyms: be owned by, be the property of, be the possession of, be in
the ownership of, be held by, be at the disposal of, be in the
hands of

The mirror is located in the room assigned to Harry. It's not exactly Harry's property, he hasn't bought nor rented the room but we say that it's his room. In the same way, the mirror placed in that room is Harry's mirror, his mirror.

I'd like to give a different take on this. The OED defines mirror as both a reflective surface, and:

A person or thing embodying a feature or characteristic deserving imitation; a pattern; an exemplar.

(For example: "In the eyes of Victoria he was the mirror of manly beauty.")

In this case, Harry's reflection is a mirror of Harry himself. Since Harry is the one causing the reflection, you could call it "his mirror" just as you would say "his shadow" or "his footprints."

So "his mirror" is just another way of saying "his doppelganger," "his double" or in this case, "his reflection."