The question type in Stack Exchange API has a field to identify if a question is bookmarked/favorited by the user, but I don't see the same field in the answer type.
Does Stack Exchange API not provide this information for an answer?
I would like to use the /questions/{ids}/answers API to get answers along with the favorite field for each answer. I don't see this option. Is this possible?
Not possible
Originally the feature was called "favorites" only for questions. It allowed people to mark a question and look at it later. The favorite count for a question public as well as the favorites of users.
This was later renamed to "bookmarks".
The change was just cosmetic - the term "favorite" tripped up some people and made them think it is some sort of endorsement. While "bookmark" was more descriptive for the usage - users could save this post for later. There should be no implied positive or negative connotation.
Finally, the feature was renamed again to "saves" and expanded - allowed users to also save answers, as well as questions and organise them into lists.
Most importantly is that the saves were made private. Unlike favorites/bookmarks you can neither see the count on a post, nor see the saved posts in a user's profile.
There is no way to get the data for which post is saved or not. This data is private by design.
The favorite_count field in the API (and FavouriteCount in SEDE) are legacy fields from the favorite/bookmark period of the feature. They are deprecated since that feature turned into saves.