Candle press

In this version of “The Story of the White Pet”, the cat sleeps in the candle press. What is the nature of this candle press? Are we talking about a big tub?

Candle press is to candles as linen press is to linens?
(A cabinet to store candles in?)

You must be right; right afterwards it says the thief “put his hand in the box”. So, we know it is some kind of boxy enclosed storage space big enough for a cat to sleep in.

This is correct. It is a box where candles are stored. Rather like a linen press keeps the linens from getting dirty and eaten by bugs, a candle press keeps candles from getting dirty and melty.