This object could have been many of the guesses. And maybe it has been each of them over the years. A hundred years plus can take you far and wide. You and I both know that re-purposing is often in the cards for old things. Or as some like to say old things are given a new life with a new purpose.
I have looked up most of your comments on the internet to find similar looking items that match your descriptions here are some of them:
Many of you guessed as I did that it was an incense burner.
An antique hair receivers. Hair receivers kept the hair from one's hair brushings. Hair receivers had a hole on top to stuff the hair "rats" inside. With the "rats" as Kathie reminded me, were kept to make a bun larger. Or to add a curl in a special place.
As Laurie said,
"Maybe a box to hold the hair cleaned from your brush. My grandmother used a little porcelain box that looked something like this-not as big though. She was born in 1899 and I guess that was something they did back in the day. Maybe because they used these clumps of hair for adding poof to different styles or for springtime bird nesting-nothing went to waste."
To see this one and many more look at this site on Pinterest: