Chicken – Reducing fat content in thick chicken stock

See Stock vs Broth - What's the difference in usage?

Like most cooking words there is no global definition. Fat content varies by ingredients and recipe. Some fat is retained in suspension, and some will be chemically attracted to components of the stock and be difficult to remove

Some people stir in the fat, most people skim it off

Commercial stock may have been centrifuged to get the fat content down to zero. This is not that practical to do at home, but can be done

Yes, it is really chicken fat rendered during the stock making process.

Called schmaltz in Yiddish, it is an ingredient in its own right. For example, you can use it to fry foods, or instead of butter in creating a roux, when you would like the chickeny flavor it provides. It is a key ingredient in matzo balls, and similarly, makes spectacularly good dumplings of various sorts. You can refrigerate it for several months, tightly covered.

It is not necessary to return it to the stock, although you may use it as an additional ingredient in the dishes you make with the stock.

Stocks are normally defatted anyway, so you do not need to return it to your stock.