1. His favourite animal to hunt was deer.

What kind of construction is it? What would be the equivalent expression for the same?

2. He loved to hunt deer.

Does (2) convey the same as (1)?
For #2 to be the same in meaning as #1, you would have to say "He loved to hunt deer more than [he liked to hunt] any other animal." It wasn't just something he liked to do, but it was his preference over other things to hunt.
to hunt is an infinitive used adjectivally.

His favourite animal (concerning what animals he liked to hunt) was deer.

It means
He loved to hunt deer more than he loved to hunt any other animals.

gives the meaning of more than any other.

He loved to hunt deer has a similar meaning, but not the exact same meaning, as you can see.