Is the money that you request from your mother—while not being in need of it—and that is given to you based on your request, not with the intention of a gift, considered among the forbidden gifts, regardless of its value, and how can one differentiate between the two situations?

If by "forbidden gift" you mean preferring some children over others, then the money your mother gives you upon your request is a gift (hiba), and she must equalize between you and your siblings. This is because the mother is like the father in prohibiting favoritism. However, favoritism is permissible if you have a need that necessitates it, or if all your siblings agree willingly. As for trivial things that people usually overlook, equalization is not obligatory in them.